The Limits of a Function

Definition of Limits:

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.

Direct Substitution Property

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

 

Example:

Evaluate the following limits

Solution:

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.

Technique 1: Factoring

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.

Technique 2: If there are fractions within fractions, try to combine the fractions.

Example:

Solution:

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

Technique 3: If there is a square root, try to multiply by the conjugate

Example:

Solution:

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

Videos

What is a Limit? Basic Idea of Limits
Basic Idea of Limits and what it means to calculate a limit.

Calculating a Limit By Factoring and Cancelling

Calculating a Limit by Getting a Common Denominator

Calculating a Limit by Expanding and Simplifying

Calculating a Limit by Multiplying by a Conjugate

Custom Search