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In these lessons, we will learn

- 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

We have also included a limits calculator at the end of this lesson. This math tool will show you the steps to find the limits of a given function.

Related Topics: More Lessons on Calculus

We write

and say “the limit of

f(x), asxapproachesa, equalsL”if we can make the values of

f(x) arbitrarily close toLby takingxto be sufficiently close toa(on either side ofa) but not equal toa.

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*.

What is a Limit? Basic Idea of Limits

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

If

fis a polynomial or a rational function andais the domain off, then

**Example:**

Evaluate the following limits

**Solution:**

How to calculate the limit of a function using substitution.

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.

**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.