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

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Lesson Parent:

Problem:

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

Answer:

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

Solution:

Many will say that the answer to this problem is that

and since we know the limit of a constant we have

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

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

$1$ .