Question

find the following limit or state that it does not exist. assume b is a fixed real number.

\\( \lim_{x \to b} \frac{(x - b)^{40} - 4x + 4b}{x - b} \\)

simplify the given limit.
\\( \lim_{x \to b} \frac{(x - b)^{40} - 4x + 4b}{x - b} = \lim_{x \to b} \left( (x - b)^{39} - 4 \
ight) \\) (simplify your answer.)

evaluate 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 b} \frac{(x - b)^{40} - 4x + 4b}{x - b} = \square \\) (type an exact answer.)
\\( \bigcirc \\) b. the limit does not exist.

Explanation:

Step1: Substitute \( x = b \) into \( (x - b)^{39}-4 \)

When \( x \to b \), \( x - b \to 0 \). So substitute \( x = b \) into \( (x - b)^{39}-4 \), we get \( (b - b)^{39}-4 \).

Step2: Calculate the value

\( (b - b)^{39}-4 = 0^{39}-4=- 4 \)

Answer:

A. \(\lim\limits_{x\to b}\frac{(x - b)^{40}-4x + 4b}{x - b}=-4\)