use the definition of continuity to determine whether f is continuous at a.\n f(x)=\begin{cases}13 - x&…

use the definition of continuity to determine whether f is continuous at a.\n f(x)=\begin{cases}13 - x& \text{if }x < 13\\0& \text{if }x = 13\\x^{2}-169& \text{if }x>13end{cases}quad a = 13\nis f continuous at 13? select the correct choice below and, if necessary, fill in the answer boxes to complete your answer choice.\na. no, because f(13) does not exist.\nb. yes, because f(13)= and (lim_{x\rightarrow13}f(x)=), and (f(13)=lim_{x\rightarrow13}f(x))\nc. yes, because f(13) and (lim_{x\rightarrow13}f(x)) exist\nd. no, because f(13)= and (lim_{x\rightarrow13}f(x)=), and (f(13)\neqlim_{x\rightarrow13}f(x))

use the definition of continuity to determine whether f is continuous at a.\n f(x)=\begin{cases}13 - x& \text{if }x < 13\\0& \text{if }x = 13\\x^{2}-169& \text{if }x>13end{cases}quad a = 13\nis f continuous at 13? select the correct choice below and, if necessary, fill in the answer boxes to complete your answer choice.\na. no, because f(13) does not exist.\nb. yes, because f(13)= and (lim_{x\rightarrow13}f(x)=), and (f(13)=lim_{x\rightarrow13}f(x))\nc. yes, because f(13) and (lim_{x\rightarrow13}f(x)) exist\nd. no, because f(13)= and (lim_{x\rightarrow13}f(x)=), and (f(13)\neqlim_{x\rightarrow13}f(x))

Answer

Explanation:

Step1: Find f(13)

Given (f(x)=\begin{cases}13 - x, &x<13\0, &x = 13\x^{2}-169, &x>13\end{cases}), when (x = 13), (f(13)=0)

Step2: Find (\lim_{x\rightarrow13^{-}}f(x))

For (x\rightarrow13^{-}), (f(x)=13 - x). Then (\lim_{x\rightarrow13^{-}}f(x)=\lim_{x\rightarrow13^{-}}(13 - x)=13-13 = 0)

Step3: Find (\lim_{x\rightarrow13^{+}}f(x))

For (x\rightarrow13^{+}), (f(x)=x^{2}-169). Then (\lim_{x\rightarrow13^{+}}f(x)=\lim_{x\rightarrow13^{+}}(x^{2}-169)=13^{2}-169=169 - 169=0)

Step4: Check the continuity condition

Since (f(13) = 0), (\lim_{x\rightarrow13^{-}}f(x)=0), (\lim_{x\rightarrow13^{+}}f(x)=0), and (\lim_{x\rightarrow13}f(x)=\lim_{x\rightarrow13^{-}}f(x)=\lim_{x\rightarrow13^{+}}f(x)=0), we have (f(13)=\lim_{x\rightarrow13}f(x))

Answer:

B. Yes, because (f(13)=0) and (\lim_{x\rightarrow13}f(x)=0), and (f(13)=\lim_{x\rightarrow13}f(x))