Question

find each limit. use -∞ or ∞ when appropriate.
f(x)=\frac{7x - 7}{(x - 7)^4}
(a) \lim_{x\to7^{-}}f(x)
(b) \lim_{x\to7^{+}}f(x)
(c) \lim_{x\to7}f(x)
(a) select the correct choice below and, if necessary, fill in the answer box to complete your choice.
○a. \lim_{x\to7^{-}}f(x)= (simplify your answer.)
○b. the limit does not exist.

Explanation:

Step1: Analyze left - hand limit

Let \(x\to7^{-}\), then \(x - 7\to0^{-}\). So \((x - 7)^{4}\to0^{+}\) (since it's an even - power), and \(7x-7\to7\times7 - 7=42\).

Step2: Calculate the limit

\(\lim_{x\to7^{-}}\frac{7x - 7}{(x - 7)^{4}}=\frac{42}{0^{+}}=\infty\)

Step3: Analyze right - hand limit

Let \(x\to7^{+}\), then \(x - 7\to0^{+}\), \((x - 7)^{4}\to0^{+}\), and \(7x - 7\to42\).

Step4: Calculate the right - hand limit

\(\lim_{x\to7^{+}}\frac{7x - 7}{(x - 7)^{4}}=\frac{42}{0^{+}}=\infty\)

Step5: Analyze the two - sided limit

Since \(\lim_{x\to7^{-}}f(x)=\lim_{x\to7^{+}}f(x)=\infty\), \(\lim_{x\to7}f(x)=\infty\)

Answer:

(A) A. \(\lim_{x\to7^{-}}f(x)=\infty\)
(B) \(\lim_{x\to7^{+}}f(x)=\infty\)
(C) \(\lim_{x\to7}f(x)=\infty\)