find the limits in a) through c) below for the function $f(x)=\frac{x^{2}-8x + 7}{x + 7}$. use $-infty$ and…

find the limits in a) through c) below for the function $f(x)=\frac{x^{2}-8x + 7}{x + 7}$. use $-infty$ and $infty$ when appropriate.\na) select the correct choice below and fill in any answer boxes in your choice\na. $lim_{x\rightarrow - 7^{-}}f(x)=square$ (simplify your answer)\nb. the limit does not exist and is neither $-infty$ nor $infty$.\nb) select the correct choice below and fill in any answer boxes in your choice\na. $lim_{x\rightarrow - 7^{+}}f(x)=square$ (simplify your answer)\nb. the limit does not exist and is neither $-infty$ nor $infty$.\nc) select the correct choice below and fill in any answer boxes in your choice\na. $lim_{x\rightarrow - 7}f(x)=square$ (simplify your answer)\nb. the limit does not exist and is neither $-infty$ nor $infty$.

find the limits in a) through c) below for the function $f(x)=\frac{x^{2}-8x + 7}{x + 7}$. use $-infty$ and $infty$ when appropriate.\na) select the correct choice below and fill in any answer boxes in your choice\na. $lim_{x\rightarrow - 7^{-}}f(x)=square$ (simplify your answer)\nb. the limit does not exist and is neither $-infty$ nor $infty$.\nb) select the correct choice below and fill in any answer boxes in your choice\na. $lim_{x\rightarrow - 7^{+}}f(x)=square$ (simplify your answer)\nb. the limit does not exist and is neither $-infty$ nor $infty$.\nc) select the correct choice below and fill in any answer boxes in your choice\na. $lim_{x\rightarrow - 7}f(x)=square$ (simplify your answer)\nb. the limit does not exist and is neither $-infty$ nor $infty$.

Answer

Explanation:

Step1: Factor the numerator

We factor (x^{2}-8x + 7=(x - 1)(x - 7)). So (f(x)=\frac{(x - 1)(x - 7)}{x + 7}).

Step2: Analyze the left - hand limit as (x\to - 7^{-})

As (x\to - 7^{-}), the numerator ((x - 1)(x - 7)\to(-7 - 1)(-7 - 7)=(-8)\times(-14)=112) and the denominator (x + 7\to0^{-}). So (\lim_{x\to - 7^{-}}f(x)=-\infty).

Step3: Analyze the right - hand limit as (x\to - 7^{+})

As (x\to - 7^{+}), the numerator ((x - 1)(x - 7)\to(-7 - 1)(-7 - 7)=112) and the denominator (x + 7\to0^{+}). So (\lim_{x\to - 7^{+}}f(x)=\infty).

Step4: Analyze the two - sided limit as (x\to - 7)

Since (\lim_{x\to - 7^{-}}f(x)=-\infty) and (\lim_{x\to - 7^{+}}f(x)=\infty), the two - sided limit (\lim_{x\to - 7}f(x)) does not exist and is neither (-\infty) nor (\infty).

Answer:

a) A. (\lim_{x\to - 7^{-}}f(x)=-\infty) b) A. (\lim_{x\to - 7^{+}}f(x)=\infty) c) B. The limit does not exist and is neither (-\infty) nor (\infty)