Question

1 fill in the blank 1 point if f is a function such that lim(x→2) (f(x)-f(2))/(x - 2)=0, which of the following must be true? (a) the limit of f(x) as x approaches 2 does not exist. (b) f is not defined at x = 2. (c) the derivative of f at x = 2 is 0. (d) f is continuous at x = 0. (e) f(2)=0 answer: type your answer... 2 fill in the blank 1 point lim(x→e) ((x³)-(e³))/(x - e) is (a) 0 (b) 3e² (c) e³ (d) does not exist answer: type your answer...

Explanation:

Step1: Recall derivative definition

The limit $\lim_{x
ightarrow2}\frac{f(x)-f(2)}{x - 2}$ is the definition of the derivative of $f(x)$ at $x = 2$. Given it equals 0, the derivative of $f$ at $x=2$ is 0.

Step2: Analyze limit in second - question

For $\lim_{x
ightarrow e}\frac{x^{3}-e^{3}}{x - e}$, by the formula $a^{3}-b^{3}=(a - b)(a^{2}+ab + b^{2})$, we have $\lim_{x
ightarrow e}\frac{(x - e)(x^{2}+xe+e^{2})}{x - e}=\lim_{x
ightarrow e}(x^{2}+xe + e^{2})=3e^{2}$.

Answer:

1. C. The derivative of $f$ at $x = 2$ is 0.
2. B. $3e^{2}$