Question

classify the discontinuities in the function below at the given point. f(x) = \frac{x^{2}-12x + 35}{x - 5};x = 5 select the correct choice below, and, if necessary, fill in the answer box to complete your choice. a. the discontinuity at x = 5 is a removable discontinuity. the function can be redefined at this point so that f(5)=. b. the discontinuity at x = 5 is an infinite discontinuity. c. the discontinuity at x = 5 is a jump discontinuity.

Explanation:

Step1: Factor the numerator

Factor $x^{2}-12x + 35$ as $(x - 5)(x - 7)$. So $f(x)=\frac{(x - 5)(x - 7)}{x - 5}$.

Step2: Simplify the function

Cancel out the common factor $(x - 5)$ (for $x
eq5$), we get $f(x)=x - 7$ for $x
eq5$.

Step3: Find the limit as $x$ approaches 5

$\lim_{x
ightarrow5}f(x)=\lim_{x
ightarrow5}(x - 7)=5-7=- 2$.

Since the limit exists as $x$ approaches 5 but the function is not defined at $x = 5$, the discontinuity is removable. And if we re - define the function, $f(5)=-2$.

Answer:

A. The discontinuity at $x = 5$ is a removable discontinuity. The function can be redefined at this point so that $f(5)=-2$.