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Programmers and Teachers:

# Limit Calculator

## Lesson on Limits

#### Lesson Contents

### What is a Limit?

Sometimes a function f(x) will be undefined at certain x values. Since the function is not defined at those locations, it can be hard to determine the function’s behavior. For example, f(x) = ^{1}⁄_{x} is undefined at x = 0 since ^{1}⁄_{0} is undefined.

However, we can take the limit of f(x) at x = 0. **Taking the limit of a function helps to describe its behavior around a point.** The limit is defined as the value that a function f(x) approaches as *x* approaches the point *a*. Limit notation is given as:

Where *L* is the limit of *f(x)* as *x* approaches *a*. This notation describes a two-sided limit, where *x* approaches *a* from both sides. We can take a left-hand or right-hand limit where *x* approaches from one side by adding a minus sign (for left-hand limit, ex: x → a^{–}) or a plus sign (for a right-hand limit, ex: x → a^{+}) next to the *a*.

### How to Hand Calculate a Limit

To take the limit of a function, we plug in numbers very close to the *x* value we are approaching. Plugging in incrementally closer numbers gives us an idea of what the function *would* be if it were not undefined at that point.

As an example, let’s find the left-hand limit of f(x) = ^{1}⁄_{x} as *x* approaches 0. Since x → 0^{–}, we will start by plugging in numbers that are barely under 0.

1.) f(-0.1) = ^{1}⁄_{-0.1} = -10

2.) f(-0.001) = ^{1}⁄_{-0.001} = -1000

3.) We can see that approaching 0 from the left side causes the function to become a larger negative value. The function’s value is approaching negative infinity. Therefore, the limit is -∞.

## How the Calculator Works

The calculator on this page uses a computer algebra system (CAS) to symbolically compute your limit. The computer algebra system follows the same algebra and calculus rules that we use when calculating by hand with a paper and pencil.

Since the software treats the function and numbers as *symbols*, it can calculate a perfectly accurate solution in some cases because it avoids computer round-off error. The symbolic nature of it also means it can calculate limits that approach oo and -oo.

Once the limit is computed, its value is sent back to this page. The answer is formatted in limit notation with the markup language *LaTeX* which is then rendered from text into the image that you actually see.