Question

find the limits in a), b), and c) below for the function f(x) = \frac{5x}{x - 8}. use -\infty and \infty when appropriate.
a. \lim_{x\to8^{+}} f(x)=\infty (simplify your answer.)
b. the limit does not exist and is neither -\infty nor \infty.
c) select the correct choice below and fill in any answer boxes in your choice.
a. \lim_{x\to8} f(x)= (simplify your answer.)
b. the limit does not exist and is neither -\infty nor \infty.

Explanation:

Step1: Analyze right - hand limit

As \(x\to8^{+}\), the numerator \(5x\to5\times8 = 40\) and the denominator \(x - 8\to0^{+}\). So \(\lim_{x\to8^{+}}\frac{5x}{x - 8}=\infty\).

Step2: Analyze two - sided limit

For \(\lim_{x\to8}\frac{5x}{x - 8}\), as \(x\to8^{+}\), \(\lim_{x\to8^{+}}\frac{5x}{x - 8}=\infty\), and as \(x\to8^{-}\), the numerator \(5x\to40\) and the denominator \(x - 8\to0^{-}\), so \(\lim_{x\to8^{-}}\frac{5x}{x - 8}=-\infty\). Since the left - hand limit \(\lim_{x\to8^{-}}\frac{5x}{x - 8}\) and the right - hand limit \(\lim_{x\to8^{+}}\frac{5x}{x - 8}\) are not equal.

Answer:

A. \(\lim_{x\to8^{+}}f(x)=\infty\)
B. The limit does not exist and is neither \(-\infty\) nor \(\infty\)