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Limits

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How to Change Unacceptable Forms before Using L'Hôpital’s Rule

You can use L'Hôpital’s rule to find a limit for which direct substitution doesn't work. If substitution produces one of the unacceptable forms, ± ∞ · 0 or ∞ – ∞, you first have to tweak the problem to [more…]

How to Solve Limits by Conjugate Multiplication

To solve certain limit problems, you’ll need the conjugate multiplication technique. When substitution doesn’t work in the original function — usually because of a hole in the function — you can use conjugate [more…]

How to Solve Limits at Infinity by Using Horizontal Asymptotes

Horizontal asymptotes and limits at infinity always go hand in hand. You can’t have one without the other. If you’ve got a rational function like [more…]

How to Use Limits to Determine Continuity

Here you’ll learn about continuity for a bit, then go on to the connection between continuity and limits, and finally move on to the formal definition of continuity. [more…]

Solving Easy Limits

There are two types of easy limit problems: the ones you should just memorize and the ones where you can plug in the x-number and get the answer in one step. [more…]

How to Solve Limits with a Calculator

You can solve most limit problems by using your calculator. There are two basic methods. For example, say you want to evaluate the following limit: [more…]

How to Solve Limits by Factoring

You can use the algebraic technique of factoring to solve “real” limit problems. All algebraic methods involve the same basic idea. When substitution doesn’t work in the original function — usually because [more…]

How to Solve Limits with Basic Algebra

When substitution doesn’t work in the original limit function — usually because of a hole in the function — you can often use some algebra to manipulate the function until substitution does work [more…]

How to Solve Limits at Infinity with a Calculator

Solving for limits at infinity is easy to do when you use a calculator. For example, enter the below function in your calculator's graphing mode: [more…]

How to Solve Limits at Infinity by Using Algebra

Yes, you can solve a limit at infinity using a calculator, but all things being equal, it’s better to solve the problem algebraically, because then you have a mathematically airtight answer. For example [more…]

How to Calculate Instantaneous Speed with Limits

You can calculate the instantaneous speed of an object using limits. Say that you and your calculus-loving cat are hanging out one day, and you decide to drop a ball out of your second-story window. Here’s [more…]

How to Make Unacceptable Forms Acceptable before Using L'Hôpital’s Rule

You can use L'Hôpital’s rule to find a limit when other methods don’t work. In fact, even if some other method does work, L'Hôpital’s rule is often a good shortcut. If substitution of the limit number [more…]

How to Determine Limits of Sequences with L'Hôpital's Rule

You can use L’Hôpital’s rule to find limits of sequences. L'Hôpital's rule is a great shortcut for when you do limit problems. Here it is: [more…]

How to Use L'Hôpital's Rule to Solve Limit Problems

L’Hôpital’s rule is a great shortcut for doing some limit problems. (And you may need it someday to solve some improper integral problems, and also for some infinite series problems.) [more…]

How to Solve Limits with a Limit Sandwich

When you can't solve a limit by using algebra, try making a limit sandwich. The best way to understand the sandwich, or squeeze, method is by looking at a graph. [more…]

Evaluating Limits in Calculus

The mathematics of limits underlies all of calculus. Limits sort of enable you to zoom in on the graph of a curve — further and further — until it becomes straight. Once it's straight, you can analyze [more…]

How to Determine Whether a Function Is Discontinuous

As your pre-calculus teacher will tell you, functions that aren't continuous at an x value either have a removable discontinuity (a hole in the graph of the function) or a [more…]

How Limits Work with Functions

Not every function is defined at every value of x.Rational functions, for example, are undefined if the denominator of the function is 0. You can use a [more…]

How to Estimate the Limit of a Function Analytically

If your pre-calculus teacher asks you to estimate the limit of a function analytically, you can simply set up a chart and put the number that x is approaching smack dab in the middle of it. Then, coming [more…]

How to Find the Limit of a Function Algebraically

When your pre-calculus teacher asks you to find the limit of a function algebraically, you have four techniques to choose from: plugging in the x value, factoring, rationalizing the numerator, and finding [more…]

How to Use the Limit Laws in Calculus

If you know the limit laws in calculus, you'll be able to find limits of all the crazy functions that your pre-calculus teacher can throw your way. Thanks to limit laws, for instance, you can find the [more…]

How to Determine Whether a Function Is Continuous

A graph for a function that's smooth without any holes, jumps, or asymptotes is called continuous. Your pre-calculus teacher will tell you that three things have to be true for a function to be continuous [more…]

How to Find the Limit of a Function Graphically

When you’re given the graph of a function and your pre-calculus teacher asks you to find the limit, you read values from the graph — something you’ve been doing ever since you learned what a graph was! [more…]

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