![]() ![]() Leading: When each round begins, the player with the two of clubs leads that card to begin the first trick of the round.This pattern repeats with subsequent rounds. You keep the cards dealt to you in the fourth round. The person you pass to is determined by what round you’re playing-to the person on your left in the first round, to the person across from you in the second round, and to the person to your right in the third round. Passing: You get to pass three cards to another player at the beginning of every three out of four rounds.You play as many rounds as needed until one player reaches 100. So each player begins each round with 13 cards. Rounds: You play hearts in rounds, and in each round, all four players are dealt the entire deck of cards.Trick-taking games, like Hearts, usually have unique components, and we break down the important aspects of this game in the following list: To do this, avoid taking tricks with penalty cards unless you are trying to shoot the moon. ObjectiveĮnd the game with the lowest score. ![]() On Solitaired, you can play unlimited free games of Hearts. You play rounds until someone reaches 100 points, and the winner is the player with the lowest score. Within each round are 13 tricks, and points are given to players who take tricks containing heart cards (1 point each) and tricks containing the queen of spades (13 points). Hearts is a trick-taking card game, typically played with four players, where the goal is to end the game with the fewest points. ![]()
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![]() ![]() How to Change the Default Save Location for new Files and new Modern Apps from Storage Settings. The "Move" feature doesn't work anymore, either for the pre-installed Modern Apps (default apps) and the installed (by you) Apps. At the right side, select the App that you want to move to another location and then click the Move button. Go to Settings > System and click Apps & features on the left side.Ģ. In order to move the already installed apps to another drive:ġ. If you want to change the default location for new apps, see method 2 below: Notice 1: Pre-installed Apps cannot be moved to another location (drive). Change/Move the Default Location for installed Modern Apps. From now on all your files can be found at the new location and all the new files will be stored at same location too. Finally press OK to exit the folder's Properties. Click Apply and then click Yes to confirm your decision.ħ. ![]() *Į.g.: In this example I have chosen to move the contents of "Documents" folder, from their default location "C:\Users\Admin\Documents" to "F:\My Files\Documents" folder.Ħ. Choose the new destination and click Select folder. to another drive) press the Move button.ĥ. To move all the files from the current location to a another location (e.g. * For example: If you have selected the "Documents" folder, then you will see where all your documents are currently stored. – Here you can see the current Save location of the selected folder. Select the folder that you want to move its contents to another drive (e.g. * Note: In Windows 7 and Vista you have to navigate to "C:\Users\%Your_User_Name%" folder.Ģ. Open Windows Explorer and double-click at This PC at the left side. To move Personal Folders location to another location and change the default Save location for new Files to the new location:ġ. Notice-2: This method applies also to Windows 8, 7 and Vista. Notice-1: By using this procedure you will move your existing personal files (Documents, Music, Pictures, Videos, etc.) to a new location and change the default save location for new files. How to Move Personal Folders and Modern Apps to another location. ![]() Change the Default Location of your Personal Folders by using Registry. Change the Default Save Location for New Apps and New Files by using Storage settings. Move Personal Folders and Modern Apps to another location (Drive). How to Change the default Save location for Personal Folders and Modern Apps. In this tutorial you can find detailed instructions on how to move or change the default location of your Personal Folders/Files and Modern Apps to another location on your computer. ![]() Important: The procedure of moving your personal files and folders to another drive is essential if your main disk does not have enough free space for the proper Windows operation. By default all users' files (Personal Files) on Windows Vista and later Windows OS are saved on several Personal folders (Documents, Pictures, Music, Videos, etc.) under the "C:\Users\%Username%\" folder, which is located on the same partition (drive) where the operating system's files are also located (system disk).īut sometimes if the storage needs exceed the disk's limits and we have a secondary drive with enough disk space on our system, then it is better to move all our personal files and the default Save Location for the newly created files to the secondary drive in order to free up space on the system disk. ![]() ![]() ![]() This free spreadsheet includes our very popular budgeting guidelines so that you have an idea about how your spending compares to that of other Canadians with similar sized households. There’s no such thing as a “normal” budget, but based on tens of thousands of credit counselling appointments, we’ve been able to create Canadian guidelines for how people will generally want to spend their after-tax income to avoid getting into debt.Įveryone allocates their money for personal or household expenses and savings a bit differently. What Does a Normal Personal or Household Budget Look Like?īudgeting and money management guidelines for after-tax income and to avoid debt Easy to use with more personalized expense categories than other similar templates.If you want to reduce your expenses, it can suggest dozens of places where you could consider reducing your expenses to save money.Identifies common budgeting pitfalls and offers tips and solutions to fix them.Keeps a running tally of how much you’re planning to spend versus your income.Drop down menus let you tell it whether each expense occurs weekly, monthly, annually, etc.Stand-alone Excel or Open Office file keeps your information private on your own computer. ![]() Clear graphs, dollar values and percentages so that you can adjust things as you go.This allows you to compare your spending to what's "normal" Provides guidelines for how much to spend in each category of your budget.Simple, helpful graphs let you see your monthly spending in a whole new way (just in case you're not a numbers person).User-friendly, even if you don’t really know how to use Excel.A calculator spreadsheet that is smart and interactive, to guide you through the budgeting process. ![]() Our free budgeting worksheet will help you manage your money better. You can also download the calculator spreadsheet on the Mac and open it in Numbers:ĭownload Mac version as Numbers Spreadsheet (.numbers) File If you don't have Microsoft Excel, you can download OpenOffice for free and then download the file below:ĭownload Open Office Spreadsheet (.ods) File If you have doubts about how much to spend on certain expenses, aren’t sure if you did your budget right, or want help planning your monthly budget so that you can manage money better, this is the free Excel budget calculator spreadsheet to download now! ( the Mac version works great too)įeatures of Our Free Budget Worksheet and Calculatorĭownload Microsoft Office Excel 2007 Worksheet (.xlsx) Fileĭownload Microsoft Office Excel 97-2003 Worksheet (.xls) File Many people put off creating a household budget because they think that they don’t really know how to budget. To make budgeting easier for you and your spouse, family or household, we’ve created an intelligent, interactive Canadian budget calculator spreadsheet in Excel that you can download and use for free as your personal budget template (it's now available on the Mac too). Free Interactive Budget Calculator Spreadsheet for Canadians – Easy to Use Excel Template to Help You Create a Personal Budget ![]() ![]() ![]() ![]() With the amount of television, internet and social media usage these days this tool couldn't be more relevant! Great for: This coaching exercise is very versatile and helps absolutely everyone better understand how they currently spend their time - and to identify and eliminate their unique time-wasters. So like "Big Rocks" it's a great tool for business and corporate use. This coaching tool helps people become very AWARE of how they could waste less time - and be more productive. Great for: Anyone who struggles with many priorities and tasks - especially moms! Also great for business, executive, career coaches and corporate workshops to help with procrastination, improving productivity and being more focused. You coach them around identifying, learning to prioritise and action the "Big Rocks" first, and then to put their metaphorical pebbles and sand in AFTERwards. ![]() The story and concept comes first and then a jar with "Big Rocks" is used as a visual to help them prioritise their activities and cement the learning. The "Big Rocks" Coaching Exercise uses a powerful metaphor of trying to fit rocks, pebbles and sand in a jar and how we tend to fill our lives with sand and pebbles first and leave no room for our "Big Rocks".Great for: life-coaching, spiritual workshops, one-on-one sessions and also for homework as part of values exploration. Ultimately this exercise helps our clients connect with what makes them truly happy and to see how easy it can be to create that. Then you coach them to see how the experiences they've chosen are both easy to create and - usually - free! Great for helping people begin to understand what creates joy and happiness. "What Makes My Heart Sing?" This coaching exercise draws out the experiences where people are in flow or very present with themselves.Also see our article with 12 different ways you can use the Wheel of Life here. It's endlessly usable for different applications. Recently updated, I consider this to be the ultimate coaching tool. FREE: "The Wheel of Life" Coaching Exercise.So as you're reading this, if you think of any essential coaching tools you use, simply comment below! Here are the 9 Coaching Exercises and Templates I would not be without: First, if you're wondering what Coaching Exercises are, read our Complete Guide to Coaching Exercises & Tools >Īnd we'd also love to hear from you which tools and exercises you use most - or find most useful - in your practices. Below, I share the tools that I think are most universally useful for coaches and our clients, plus a little on how to use them! Enjoy. I am often asked which Coaching Tools and Exercises I like best, use most, would recommend or would not be without. ![]() ![]() Anime games also boast exceptional soundtracks, with music that complements the narrative and elevates the gaming experience. The strategy genre melds seamlessly with the rich storytelling that anime is known for, offering engaging gameplay experiences. Strategy enthusiasts can test their tactical prowess in anime-themed strategy games, where they command armies or lead heroes into battle. These games often introduce players to memorable characters and heartwarming, or heart-wrenching, tales. For those who yearn for emotional depth and narrative complexity, dating simulators and visual novels offer intricate storylines where players' choices influence the outcomes of romantic entanglements and dramatic narratives. Action aficionados can step into the shoes of their beloved characters from iconic series, engaging in thrilling battles and epic adventures. There's a game for everyone, regardless of their preferences. Within the realm of anime games, diversity reigns supreme. Whether it's the intense action of shonen anime or the tender romance of shoujo manga, these games faithfully encapsulate the essence of anime. This meticulous attention to detail brings these virtual worlds to life, captivating players with stunning visuals, vibrant colors, and meticulously crafted character designs. One of the most striking features of anime games is their commitment to replicating the unique and captivating art styles found in anime series and films. Anime Games are mostly free fighting and dress up games featuring Japanese cartoon characters. These games are a tribute to the rich world of anime, catering to fans of various genres and styles found in this cherished medium. ![]() ![]() ![]() ![]() ![]() Others may not like to load their Fridays with heavy tasks because the likelihood that you can’t complete or follow up on a Friday task is high. Divide your week into more and less structured daysĪ good rule of thumb is to divide your week into the days when you are most productive and days when you need a little more freedom in the schedule.įor instance, some people like to jump into Mondays head-first because their energy and enthusiasm are high. Simply downloading everything you do during the week onto a calendar may demonstrate how busy you are, but it won't help you attain any progress in your week. Limit your scope to what is achievable in a short windowĬreating a weekly planner should be structured and calculated.Define a strategy for how you’ll accomplish your work.Get aligned with teammates about work load and responsibilities.Visualize all the work you have for the week.This can be especially useful to have a place to plan your time at work without having to create events in your official work calendar. Task ID: Provide a unique identifying number for each task on the project for easy reference.Use this framework to plan a week.Stage or Phase: Break your project timeline down into stages that include key tasks for each phase of the project.Sprint: For Agile teams, clearly label each sprint on your project timeline so that team members can see which sprint-specific tasks must be completed within a specific timeboxed period.This includes all aspects of the project, including tasks, activities, departments, timeline overview, key deliverables, and milestones. Project Scope: Provide a brief description of what is in scope for the project.Objectives or Goals: List the project’s key objectives, such as the problem it will address and what goals it must meet by project completion.Department: Include the name of any department responsible for a project task or phase, or the department on which a task or project phase has a dependency.At-Risk Tasks: Identify tasks that are in danger of not being completed, and notify relevant team members.Milestone: Define your project’s time-sensitive milestones that must be met in order for the project to be successfully completed.Īdditionally, some project timeline templates include the following components:. ![]() Status: Select a status for each individual project task (e.g., Not Started, In Progress, On Hold, or Complete).Duration: Review the task duration, which is calculated by each task’s defined start and end dates.End Date: Set the date for when the individual task must be completed.Start Date: Enter the date that each individual task needs to be started.Task Owner: Assign the team member who is responsible for completing each individual project task.Task Name: Provide a name for each project task that needs to be completed in order to meet the criteria for project success.Project Name: Enter the name of the project that you and your teammates will be undertaking.While project timeline templates may vary, most of them include the following components to ensure that you account for everything on your project roadmap: With its horizontal bar chart, a project timeline provides team members with a clear visual map of individual task delivery dates, priorities, and current statuses, as well as whether or not they have task dependencies. Use a project timeline template to add key project milestones so that it’s clear during project planning what primary actions or events need to be met, and when. Timeline templates also serve as a roadmap for your project so that all stakeholders are aware of project goals. Use the template for a visual representation of your project’s goals, scope, and tasks to ensure timely project delivery.Īs the primary vehicle for managing your project’s deliverables, a project timeline template provides a comprehensive method for tracking project tasks, assigning them to team members, setting delivery dates, and monitoring their progress. Getting started with the Smartsheet APIĪn Excel project timeline template is a visually dynamic tool that project managers can use to plan and monitor project progress.ENGAGE Smartsheet ENGAGE brings together our global customers, experts, and partners to share their experiences, ideas, and best practices.Smartsheet events Your hub for Smartsheet events, webinars, Q&As, and user groups.Partners Learn about the Smartsheet partner program and access our partner directory.Community Explore user-generated content and stay updated on our latest product features.Help and Learning A comprehensive knowledge base, including articles, tutorials, videos, and other resources that cover a range of topics related to using Smartsheet.Content Center Articles and guides about project management, collaboration, automation, and other topics to help you make the most of the Smartsheet platform. ![]() ![]() ![]() Once clicked or tapped, the resulting animation turns the previously colorless heart red. ![]() Each of these actions must be validated by feedback.įor example, e-retailer Etsy has a heart icon that allows users to add an item to their favorites. Product pages are all about helping users perform actions related to conversion, such as selecting item styles and quantities, adding products to the cart, or bookmarking favorite items. Dots should be hollow by default and filled to emphasize the current image. Dot or arrow controls help users identify how to move within image carousels. Navigation should be easily identifiable. Similarly, follow through is the subtle motion following the motion sequence, which helps to make the animation seem more realistic. This brief movement in the opposite direction of the main action helps build momentum and excitement. Anticipation is the motion before the main action sequence (the left/right slide animation in the image carousel). Anticipation at the beginning of the slide animation and follow through at the end will help ensure that your transitions are seamless. Smooth, continuous slide microinteractions help create the illusion of natural interaction with physical objects. Image carousels allow users to preview several products in a short time without leaving the category page.įor the best image carousel experience, Rashni recommends focusing on smooth transitions and intuitive navigation: As a user scrolls down the homepage, content appears at intervals and varying speeds, mimicking real-world parallax and acting as a form of progessive disclosure that makes it easier for the user to scan and digest information on the screen. Parallax scrolling is a powerful animation technique for establishing the visual hierarchy of homepage content, such as product photos, descriptions, and categories. Rashni explains that animation that organizes large amounts of information contributes to more intuitive design, which in turn helps to hold a user’s attention. Animation can make drop-down menus more engaging and reinforce product categorization, making it easier for users to find items and reducing drop-off rates. Homepage design choices, such as navigation, need to be strategic to capture a user’s attention. “The more realistic the experience, the deeper the connection between the user and the product, and the more likely a repeat visit to the site.” The E-commerce Homepageĭigital shoppers have dwindling attention spans and little patience for complex UIs. ![]() “For example, when the user adds an item to their cart, scale-down and drop-in-cart microinteractions make this action feel more realistic, as if the user were shopping in a brick and mortar store,” she says. “Many e-commerce UI elements have functionalities that can benefit from microinteractions in order to engage the customer and provide vital information,” says Rashni Parichha, a product designer and member of the Toptal network. For example, the search box on an e-commerce homepage might signal interactivity by expanding when a user hovers or taps on it then it may reveal animated autosuggestions as users type search queries. Animated microinteractions aid in this journey of discovery. At each step, more information is revealed. The e-commerce sales funnel has four common steps-homepage, category page, product page, and checkout-that guide users from a company’s initial value proposition to purchase. Animated UX Microinteractions in the E-commerce Sales Funnel According to UX designer Alexandre Brito, a member of the Toptal network since 2016, even simple animated effects-like the scrollbar, pull-to-refresh, and swipe-can help create intuitive digital shopping experiences. In addition to helping users find and buy products, they can foster habit loops that result in repeat customers.ĭespite their importance in the e-commerce funnel, animated microinteractions don’t need to be especially fancy. These subtle motion effects provide quick visual feedback when a user interacts with a UI component, such as a menu or button, and are meant to improve findability and guide users as they navigate between e-commerce pages. When applied throughout e-commerce experiences, animated microinteractions communicate system progress, improve user satisfaction, and increase conversion rates. ![]() ![]() ![]() ![]() But after her grandfather died and her uncle, King Edward VIII, abdicated - both within the same year - Elizabeth’s father became King George VI. Source: Hemophilia.When Elizabeth Alexandra Mary Windsor was born, she was not a direct heir to the throne since her father was the second son of King George V. This disease is carried by females and manifested by males. Two of her five daughters, Princess Alice and Princess Beatrice, passed the mutation to various royal houses across Europe. Her son Leopold died of a hemorrhage after a fall when he was 30. She passed the trait on to three of her nine children. Queen Victoria was believed to be a carrier of hemophilia B, or factor IX deficiency. Her daughter would become Victoria Eugenie, Queen of Spain. Princess Beatrice married Prince Henry of Battenberg. His granddaughter would marry the King of Sweden. Prince Leopold suffered from Hemophilia and died at the age of 30. Her descendants became the Queen of Denmark, and her descendants were Danish and Greek monarchs. His daughter Princess Margaret became Crown Princess of Sweden. His wife was Princess Louise Margaret of Prussia. He also served as Governor General of Canada. The Province of Alberta was named for the Princess. Her Husband, John Douglas Sutherland Campbell. Princess Louise Caroline married a commoner. She had 6 children from whom there are no descendants alive today. His daughter Marie became Queen of Romania. Her daughter, Alexandra or Alix, became empress consort of Russia when she married Nicholas II, the last czar of Russia. Princess Alice was the Queen Consort of Hesse and Rhine. The Norwegian monarchs descend from this line. King Edward's daughter Maud became Queen consort of Norway. His descendants include King George V, King George VI, Queen Elizabeth II, and King Charles III. Prince Albert Edward, later King Edward VII became king of England. Their son would be the last emperor of Germany. Princess Victoria married Friederich Wilhelm of Prussia. This tree also shows crowned monarchs in yellow so you can determine the extent of power Queen Victoria's descendants have reached. Otherwise, the tree would be too tangled to show important relations. Just the most important ones are included. Many cousins and second cousins married each other. ![]() The interactive family tree above does not show all connections. These connections began with her children. Her descendants became monarchs of England, Greece, Denmark, Serbia, Romania, Spain, Norway, Sweeden, and Russia. This is why Queen Victoria is called Europe's Grandmother. Queen Victoria's Descendantsįollowing with the principle that marriage was a form of power and a path to help spread British liberal values across the continent, Queen Victoria matched her children with suitable families in power. A captivating exploration of the role in which Queen Victoria exerted the most international power and influence: as a matchmaking grandmother. If you want to learn more about her descendants, Amazon has Queen Victoria's Matchmaking: The Royal Marriages That Shaped Europe. Prince Albert and Queen Victoria believed that dynastic marriages between their nine children and European royalty would provide a peaceful future for Europe. Queen Victoria understood the importance of dynastic marital alliances to extend royal influence. Prince Albert Edward, Prince of Wales, King later Edward VII.Victoria and Albert had 9 children and 42 grandchildren. Queen Victoria wore black every day after his death. He was Prince Consort until his death at 42 years old. Queen Victoria married her first cousin on her mother's side, Prince Albert of Saxe-Coburg and Gotha. This personal history is a compelling look at the complex woman whom, until now, we only thought we knew. To learn more about Queen Victoria, Amazon has Queen Victoria: A Personal History. It was also a time of extreme social inequality. It was also a period of advancement of science, railway growth, industrialization, and rapid changes. The Victorian Era was a period of great power and wealth for Britain. Victoria died on January 22, 1901, at the age of 81. She inherited the throne from her uncle, William IV, as all her uncles died without an heir.Ī few weeks after turning 18, Victoria became Queen of England on June 20, 1837, following William’s death, with the coronation taking place a year later on June 28, 1838. When Victoria was only one year old, her father died of pneumonia. Edward was the fourth son of King George III. Queen Victoria was the daughter of Edward, Duke of Kent and the German Princess Louisa Victoria. The longest ruler after Queen Elizabeth II. Queen Victoria was born at Kensington Palace on May 24, 1819. ![]() ![]() ![]() Just click & print.Ĭhoose your level and print your sudoku, you can create you own sudoku book up to 12 pages.Ĥ pages of easy sudokus, print the grids.Ĩ pages of easy sudokus, print the grids.ġ2 pages of easy sudokus, print the grids. Anything that stretches your mind will help keep your brain provides free sudokus. I have a hand-held Sudoku game (with stylus) that I play all the time.Įven though I practice, I'm not very good. Sudoku really is an excellent brain game, as it trains logical reasoning and visual perception, among other brain skills. The challenge of sudoku is using the process of elimination and other strategies to identify the unique solution for the sudoku puzzle. The numbers 1 through 9 must appear exactly once in each 3x3 box. The printable bundle consists of Sudoku puzzles which range from difficulty level easy to hard, so no matter the level of difficulty your family desires, there’s something for everyone. The numbers 1 through 9 must appear exactly once in each column. The numbers 1 through 9 must appear exactly once in each row. To figure out which number between 1 and 9 belongs in a particular cell, you need to adhere to the following rules: Your job is to determine the correct number to be entered in each of the empty cells. ![]() Some of the cells in the grid are prefilled with a number between 1 and 9, while many other cells are blank. ![]() For kids and with large print for adults. This grid is subdivided into nine 3 x 3 boxes. Free printable sudoku puzzles in a variety of sizes, including 4x4, 6圆, and difficulty levels, from easy to hard. The hard part lies in solving the puzzle!Ī sudoku puzzle consists of a 9 by 9 grid. Free printable easy sudoku puzzles,Easy puzzles are great for getting started in sudoku, or as a warm-up for tougher stuff.Play easy online sudoku on the. Here's a summary of the rules of sudoku, in case you've never played before or need a refresher. Printable Sudoku Puzzles - Rules of Sudoku The solutions pages are provided as well. Printable sudoku 4 puzzles per page, in PDF or HTML, according to 5 levels: easy, medium, hard. Compatible with all browsers, tablets and phones including iPhone, iPad and Android. To print a puzzle page, click the "Print This Page" button at the top of the page. Just click & print thousands of free sudoku puzzles 9x9. Compatible with all browsers, tablets and phones including iPhone, iPad and Android. Reprinted by Permission.") The 100 Free Sudoku Printables ![]() Your readership will love it! (In your publication, simply include the following copyright statement next to the puzzle: "© . Puzzles help keep your mind sharp while enjoying a challenging logic game. Want to play sudoku online in your browser? Visit my Free Sudoku Games page.Īre you the publisher of a newsletter, bulletin, newspaper, or magazine? Include a sudoku puzzle or word search puzzle in each issue. More Free Printables: Crosswords, Word Searches, Bible Word Searches, Special Needs Word Searches, Mormon Word Searches, Balance Quest Math Puzzles For detailed instructions, including advanced techniques, visit the Sudoku Instructions page. See below for a summary of the rules of sudoku. Or if you're a sudoku whiz, go for the harder ones. If you are new to sudoku, start with an easy puzzle. The printable sudoku puzzles pages below are listed in order of difficulty. Note: For even more sudoku, see the 100 Free LARGE Sudoku Puzzles page. There is a link to the Solutions on the puzzle page as well. On this page are 100 free sudoku puzzles I made that you can print out right now. ![]() ![]() So this is one of the, one of the trig values Square root of 2 over 2, we're assuming this is in radians, if we're thinking inĭegrees, this would be a negative 45-degree angle, Now, negative pi over 4, sine of negative pi over 4 is going to be negative square root of 2 over 2, so this is negative Going to be equal to this expression evaluatedĪt negative pi over 4, so 1 plus square root of 2 times sine of negative pi over 4, over cosine of 2 times negative pi over 4. Make sure we can see that negative there, of cosine of 2 theta, and both of these expressions are, if these were function definitions or if we were to graph y equals 1 plus square root of sine, square root of 2 times sine theta, or y equals cosine of 2 theta, we would get continuous functions, especially at theta isĮqual to negative pi over 4, so we could just substitute in. Going to be the same thing as the limit, as theta approaches negative pi over 4 of 1 plus square root of 2 sine theta over the limit as theta approaches negative pi over 4. ![]() Well, one take on it is well, let's just, let's just say that this is It a shot before we go through it together. Root of 2 sine of theta over cosine of 2 theta, as theta approaches negative pi over 4. See if we can find the limit of 1 over the square In other words, the discontinuity doesn’t really exist, it only seems to exist due to the equation not being in its simplest, i.e. If a function can be factored, and the factoring removes a discontinuity, it seems more logical to me to state that the original function had a false or phantom discontinuity. I am comfortable performing such manipulations, as it makes inherent sense to me that the limits are equivalent but I wouldn’t know how to mathematically state or prove that such manipulations are correct. I believe mathematicians refer to such a discontinuity as a ‘removable discontinuity’. where a factored equation has removed a discontinuity which was present in the original unfactored equation, and that the limits are considered equivalent. I want to better understand this distinction as Sal has identified this situation several times in multiple videos, i.e. For I learned from the video on Epsilon-Delta proofs that a limit can still exists at 'a' despite a function being undefined at 'a'. Is there a mathematical theorem or proof that states the limit of a factored equation is equivalent to the limit of the respective unfactored equation as long as the factored equation is continuous at the point for which the limit is taken? Is this distinction covered by the Epsilon-Delta proof of limits. Sal goes on to say that the limits of f(x) and g(x) are equivalent at 'a'. And that f(x) is defined and is continuous at ‘a’. I'm assuming this means that g(x) is defined for all real numbers except 'a', and therefore is not continuous at 'a'. In the video, Sal states that f(x) and g(x) are equivalent except at ‘a’, and that f(x) is continuous at ‘a’. I’m assuming that g(x) is the original unfactored equation and that f(x) is the resulting factored equation. In his explanation he speaks of two functions f(x) and g(x). To offer financial support, visit my Patreon page.8:20 in the video, Sal begins to explain that it is important to understand that the factored equation is not the same as the original unfactored equation. We are open to collaborations of all types, please contact Andy at for all enquiries. The clear explanations, strong visuals mixed with dry humor regularly get millions of views. Andymath content has a unique approach to presenting mathematics. Visit me on Youtube, Tiktok, Instagram and Facebook. ![]() ![]() In the future, I hope to add Physics and Linear Algebra content. Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. If you have any requests for additional content, please contact Andy at He will promptly add the content. \(\,\,\,\,\,\displaystyle\frac\)Ī is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. ![]() |
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