Determine \[ \lim_{x\to -2}\frac{x^2-4}{x^2-4}. \]

Primary tabs

Lesson Parent: 
Function Limits VIII: Limits Involving Zero
Problem: 

Determine
\[
\lim_{x\to -2}\frac{x^2-4}{x^2-4}.
\]

Answer: 

\[ \lim_{x\to -2}\frac{x^2-4}{x^2-4}=1 \]

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



Solution: 

Many will say that the answer to this problem is that

<Sign in to see all the formulas>

and since we know the limit of a constant we have

<Sign in to see all the formulas>

This is correct, but there is one detail that make all of this work that is important to note. Specifically, we can't have the denominator in

<Sign in to see all the formulas>

ever be zero. But this is actually taken care of for us in the definition of function limits. It is in the spot where we say "if <Sign in to see all the formulas>" and more specifically when <Sign in to see all the formulas>. Because of this <Sign in to see all the formulas> and the function limit definition does not say anything about $f(x)$ when $x=a$. Because the function limit definition does not look at the value of <Sign in to see all the formulas> when $x=-2$ the denominator is not zero and we can do the division to get $1$.