use the definition of continuity to determine whether f is continuous at a. f(x) = { (x^2 - 49)/(x - 7) if…

use the definition of continuity to determine whether f is continuous at a. f(x) = { (x^2 - 49)/(x - 7) if x≠7; 1 if x = 7; a = 7. is f continuous at 7? select the correct choice below and, if necessary, fill in the answer boxes to complete your answer choice. o a. no, because f(7)= and lim f(x)= and f(7)≠lim f(x) as x→7. o b. no, because f(7) does not exist. o c. yes, because f(7)= and lim f(x)= and f(7)=lim f(x) as x→7. o d. yes, because both f(7) and lim f(x) exist as x→7.

use the definition of continuity to determine whether f is continuous at a. f(x) = { (x^2 - 49)/(x - 7) if x≠7; 1 if x = 7; a = 7. is f continuous at 7? select the correct choice below and, if necessary, fill in the answer boxes to complete your answer choice. o a. no, because f(7)= and lim f(x)= and f(7)≠lim f(x) as x→7. o b. no, because f(7) does not exist. o c. yes, because f(7)= and lim f(x)= and f(7)=lim f(x) as x→7. o d. yes, because both f(7) and lim f(x) exist as x→7.

Answer

Explanation:

Step1: Find $f(7)$

Given $f(x)=\begin{cases}\frac{x^{2}-49}{x - 7},&x\neq7\1,&x = 7\end{cases}$, so $f(7)=1$.

Step2: Find $\lim_{x\rightarrow7}f(x)$

For $x\neq7$, $f(x)=\frac{x^{2}-49}{x - 7}=\frac{(x + 7)(x - 7)}{x - 7}=x + 7$. Then $\lim_{x\rightarrow7}f(x)=\lim_{x\rightarrow7}(x + 7)=7+7 = 14$.

Step3: Check the continuity condition

Since $f(7)=1$ and $\lim_{x\rightarrow7}f(x)=14$, and $f(7)\neq\lim_{x\rightarrow7}f(x)$.

Answer:

A. No, because $f(7)=1$ and $\lim_{x\rightarrow7}f(x)=14$ and $f(7)\neq\lim_{x\rightarrow7}f(x)$