Question

determine whether the following limit exists. compute the limit if it exists. lim(x→16) (x² - 256)/(16 - x) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim(x→16) (x² - 256)/(16 - x)=□ b. the limit does not exist.

Explanation:

Step1: Factor the numerator

We know that \(a^{2}-b^{2}=(a + b)(a - b)\), so \(x^{2}-256=(x + 16)(x - 16)\). Then the limit becomes \(\lim_{x
ightarrow16}\frac{(x + 16)(x - 16)}{16 - x}\).

Step2: Rewrite the denominator

Notice that \(16 - x=-(x - 16)\). So the limit is \(\lim_{x
ightarrow16}\frac{(x + 16)(x - 16)}{-(x - 16)}\).

Step3: Cancel out the common factor

Cancel out the \((x - 16)\) terms (since \(x
eq16\) when taking the limit), we get \(\lim_{x
ightarrow16}-(x + 16)\).

Step4: Evaluate the limit

Substitute \(x = 16\) into \(-(x + 16)\), we have \(-(16+16)=-32\).

Answer:

A. \(\lim_{x
ightarrow16}\frac{x^{2}-256}{16 - x}=-32\)