Question

determine the following limit, using ∞ or −∞ when appropriate, or state that it does not exist.
\lim_{x\to 4} \frac{4x^2 - 64}{x - 4}

simplify the expression inside the limit, if possible. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

\bigcirc a. \lim_{x\to 4} \frac{4x^2 - 64}{x - 4} = \lim_{x\to 4} (\square)
\bigcirc b. the expression inside the limit cannot be simplified.

Explanation:

Step1: Factor the numerator

First, factor the numerator \(4x^2 - 64\). We can factor out a 4 first: \(4(x^2 - 16)\). Then, \(x^2 - 16\) is a difference of squares, so it factors to \((x - 4)(x + 4)\). So the numerator becomes \(4(x - 4)(x + 4)\).
The expression is now \(\lim_{x
ightarrow 4}\frac{4(x - 4)(x + 4)}{x - 4}\).

Step2: Cancel common factors

We can cancel the common factor of \(x - 4\) (since \(x
ightarrow 4\) but \(x
eq 4\) when taking the limit, so we can cancel this factor) from the numerator and the denominator. This leaves us with \(\lim_{x
ightarrow 4}4(x + 4)\).

Answer:

For option A, the expression inside the limit after simplification is \(4(x + 4)\), so \(\lim_{x
ightarrow 4}\frac{4x^2 - 64}{x - 4}=\lim_{x
ightarrow 4}4(x + 4)\)