In these lessons, we will learn

Related Topics: More Lessons on Calculus

### Definition of Limits

Basic Idea of Limits and what it means to calculate a limit.

### Direct Substitution Property

Functions with Direct Substitution Property are called continuous at *a*. However, not all limits can be evaluated by direct substitution. The following are some other techniques that can be used.

### Factoring and Canceling

How to calculate the limit of a function by using the factorization method.

### If there are fractions within fractions, try to combine the fractions.

### If there is a square root, try to multiply by the conjugate.

### Calculating a Limit by Expanding and Simplifying

A Limits Calculator or math tool that will show the steps to work out the limits of a given function. Use it to check your answers.

You can use the Mathway widget below to practice Calculus or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

- the definition of limits
- how to evaluate limits using direct substitution
- how to evaluate limits using factoring and cancelling
- how to evaluate limits by combining fractions
- how to evaluate limits by multiplying by the conjugate
- how to evaluate limits by expanding and simplifying

Related Topics: More Lessons on Calculus

We write

and say “the limit of *f*(*x*), as *x* approaches *a*, equals *L*”

if we can make the values of *f*(*x*) arbitrarily close to *L* by taking *x* to be sufficiently close to *a* (on either side of *a*) but not equal to *a*.

This says that as *x* gets closer and closer to the number *a* (from either side of *a*) the values of *f*(*x*) get closer and closer to the number *L* In finding the limit of *f*(*x*) as *x* approaches, we never consider *x* = *a*. In fact, *f*(*x*) need not even be defined when *x* = *a*. The only thing that matters is how *f*(*x*) is defined near *a*.

Basic Idea of Limits and what it means to calculate a limit.

If *f* is a polynomial or a rational function and *a* is the domain of *f*, then

**Example:**

Evaluate the following limits

**Solution:**

**Example:**

**Solution:**

We can’t find the limit by substituting *x* = 1 because

is undefined

Instead, we need to do some preliminary algebra. We factor the numerator as a difference of squares and then cancel out the common term (*x* – 1)

Therefore,

Note: In the above example, we were able to compute the limit by replacing the function by a simpler function *g*(*x*) = *x* + 1, with the same limit. This is valid because *f*(*x*) = *g*(*x*) except when *x* = 1.

Calculating a Limit By Factoring and Canceling.

**Example:**

**Solution:**

We cannot use the substitution method because the numerator and denominator would be zero.

Calculating a Limit by Getting a Common Denominator.**Example:**

**Solution:**

We cannot use the substitution method because the numerator and denominator would be zero.

Calculating a Limit by Multiplying by a Conjugate.You can use the Mathway widget below to practice Calculus or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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