Question

let f(x) = \frac{x^{2}-3x - 10}{x - 5}. find a) \lim_{x\to5}f(x), b) \lim_{x\to0}f(x), and c) \lim_{x\to - 2}f(x).
a) select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. \lim_{x\to5}f(x)=\square (simplify your answer.)
b. the limit does not exist.

Explanation:

Step1: Factor the numerator

Factor $x^{2}-3x - 10$ as $(x - 5)(x+2)$. So $f(x)=\frac{(x - 5)(x + 2)}{x - 5}$.

Step2: Simplify the function

Cancel out the common factor $(x - 5)$ (for $x
eq5$), then $f(x)=x + 2$ for $x
eq5$.

Step3: Find $\lim_{x

ightarrow5}f(x)$
Substitute $x = 5$ into $x+2$. $\lim_{x
ightarrow5}f(x)=5 + 2=7$.

Step4: Find $\lim_{x

ightarrow0}f(x)$
Substitute $x = 0$ into $x + 2$. $\lim_{x
ightarrow0}f(x)=0+2=2$.

Step5: Find $\lim_{x

ightarrow - 2}f(x)$
Substitute $x=-2$ into $x + 2$. $\lim_{x
ightarrow - 2}f(x)=-2+2=0$.

Answer:

a) A. $\lim_{x
ightarrow5}f(x)=7$
b) $\lim_{x
ightarrow0}f(x)=2$
c) $\lim_{x
ightarrow - 2}f(x)=0$