12.2 Techniques for Evaluating Limits

Limits of Polynomial Functions 

If p is a polynomial function and c is a real number, then

Limits of Rational Functions 

If r is a rational function given by, and c  is a 

real number such that, then

,.

Evaluating Limits by Direct Substitution 

To use direct substitution:

• the function needs to be continuous.  
• there has to be no values of x that make the function undefined.
• one has to input the limit of x in all values of x to receive an output. 

Example - Evaluate:

Substitute -1 for x

Evaluating Limits by Dividing Out Technique 

When dividing out:

• Factor numerator and denominator.
• Divide out any common factors and simplify.
• Then use direct substitution for x and simplify.

Example - Evaluate: 

Factor numerator

Divide out common factor of x-5

Use direct substitution

Evaluating Limits by Rationalizing Technique 

When using the rationalizing technique:

• Rationalize the numerator of the function.
• Multiply and simplify.
• Then divide out the common factor and simply again. 
• Lastly, use direct substitution to then evaluate the limit. 

Example - Evaluate:

Rationalize the original function with:

Simplify

Divide out common factors

Then use substitution

Simplify

Existence of a Limit

If f  is a function and c and L are real numbers, then



if and only if both the left and right limits exist and are equal to L.

*sorry about the incorrectly formatted functions(the equation editor was not cooperative).