use the definition of continuity to determine whether f is continuous at a = 5. f(x)=4x + 8 is f continuous…

use the definition of continuity to determine whether f is continuous at a = 5. f(x)=4x + 8 is f continuous at 5? select the correct choice below and, if necessary, fill in the answer boxes to complete your answer choice. a. yes, because f(5)= and lim f(x)=, and f(5)=lim f(x). x→5 x→5 b. no, because f(5)= and lim f(x)=, and f(5)≠lim f(x). x→5 x→5 c. yes, because lim f(x) exists. x→5 d. no, because lim f(x) does not exist. x→5

use the definition of continuity to determine whether f is continuous at a = 5. f(x)=4x + 8 is f continuous at 5? select the correct choice below and, if necessary, fill in the answer boxes to complete your answer choice. a. yes, because f(5)= and lim f(x)=, and f(5)=lim f(x). x→5 x→5 b. no, because f(5)= and lim f(x)=, and f(5)≠lim f(x). x→5 x→5 c. yes, because lim f(x) exists. x→5 d. no, because lim f(x) does not exist. x→5

Answer

Explanation:

Step1: Calculate f(5)

Substitute x = 5 into f(x)=4x + 8. $f(5)=4\times5+8=20 + 8=28$

Step2: Calculate $\lim_{x\rightarrow5}f(x)$

Since f(x)=4x + 8 is a linear - function, $\lim_{x\rightarrow5}f(x)=\lim_{x\rightarrow5}(4x + 8)$. Using the limit rules for sums and constant - multiples of functions, $\lim_{x\rightarrow5}(4x + 8)=4\lim_{x\rightarrow5}x+\lim_{x\rightarrow5}8$. We know that $\lim_{x\rightarrow5}x = 5$ and $\lim_{x\rightarrow5}8 = 8$. So, $4\lim_{x\rightarrow5}x+\lim_{x\rightarrow5}8=4\times5+8=28$. Since $f(5)=\lim_{x\rightarrow5}f(x)=28$, the function f(x) is continuous at x = 5.

Answer:

A. Yes, because f(5)=28 and $\lim_{x\rightarrow5}f(x)=28$, and f(5)=$\lim_{x\rightarrow5}f(x)$.