$$ f ( x ) = \frac { x - 4 } { x ^ { 2 } - 16 } $$ (a) explain why ( f ) has a removable discontinuity at (…

$$ f ( x ) = \frac { x - 4 } { x ^ { 2 } - 16 } $$ (a) explain why ( f ) has a removable discontinuity at ( x = 4 ). (select all that apply.) ( lim _ { x \rightarrow 4 } f ( x ) ) is finite. ( f ( 4 ) ) is undefined. ( f ( 4 ) ) and ( lim _ { x \rightarrow 4 } f ( x ) ) are finite, but are not equal. ( lim _ { x \rightarrow 4 } f ( x ) ) does not exists. none of the above nicely done. (b) redefine ( f ( 4 ) ) so that ( f ) is continuous at ( x = 4 ) (and thus the discontinuity is remove ( f ( 4 ) = 0.125 ) exactly!

$$ f ( x ) = \frac { x - 4 } { x ^ { 2 } - 16 } $$ (a) explain why ( f ) has a removable discontinuity at ( x = 4 ). (select all that apply.) ( lim _ { x \rightarrow 4 } f ( x ) ) is finite. ( f ( 4 ) ) is undefined. ( f ( 4 ) ) and ( lim _ { x \rightarrow 4 } f ( x ) ) are finite, but are not equal. ( lim _ { x \rightarrow 4 } f ( x ) ) does not exists. none of the above nicely done. (b) redefine ( f ( 4 ) ) so that ( f ) is continuous at ( x = 4 ) (and thus the discontinuity is remove ( f ( 4 ) = 0.125 ) exactly!

Answer

Explanation:

Step1: Simplify the function

We know that (x^{2}-16=(x + 4)(x - 4)). So (f(x)=\frac{x - 4}{x^{2}-16}=\frac{x - 4}{(x + 4)(x - 4)}). For (x\neq4), we can cancel out the (x - 4) terms, and (f(x)=\frac{1}{x + 4}) when (x\neq4).

Step2: Calculate the limit

Now, we find (\lim_{x\rightarrow4}f(x)). Using the simplified form (f(x)=\frac{1}{x + 4}) (since the limit as (x\rightarrow4) does not consider the value of the function at (x = 4) itself), we substitute (x = 4) into (\frac{1}{x+4}). So (\lim_{x\rightarrow4}f(x)=\frac{1}{4 + 4}=\frac{1}{8}=0.125).

Step3: Check the function value at (x = 4)

The original function (f(x)=\frac{x - 4}{x^{2}-16}), when (x = 4), the denominator (x^{2}-16=4^{2}-16=0). So (f(4)) is undefined.

For a function to be continuous at (x=a), we need (\lim_{x\rightarrow a}f(x)=f(a)). Since (\lim_{x\rightarrow4}f(x)) is finite ((0.125)) and (f(4)) is undefined, by re - defining (f(4)=\lim_{x\rightarrow4}f(x)), we can make the function continuous at (x = 4).

Answer:

(f(4)=0.125)