Question

find the limits in a), b), and c) below for the function f(x) = \frac{7x}{x - 3}. use -\infty and \infty when appropriate.
a) \\(\lim_{x\to3^{-}}f(x)
(simplify your answer.)
o b. the limit does not exist and is neither -\infty nor \infty.
b) select the correct choice below and fill in any answer boxes in your choice.
a. \\(\lim_{x\to3^{+}}f(x)=\infty
(simplify your answer.)
o b. the limit does not exist and is neither -\infty nor \infty.
c) select the correct choice below and fill in any answer boxes in your choice.
a. \\(\lim_{x\to3}f(x)=
(simplify your answer.)
o b. the limit does not exist and is neither -\infty nor \infty.

Explanation:

Step1: Analyze left - hand limit as $x\to3^{-}$

As $x\to3^{-}$, $x - 3\to0^{-}$ (a very small negative number), and $7x\to21$. So, $\lim_{x\to3^{-}}\frac{7x}{x - 3}=-\infty$.

Step2: Analyze right - hand limit as $x\to3^{+}$

As $x\to3^{+}$, $x - 3\to0^{+}$ (a very small positive number), and $7x\to21$. So, $\lim_{x\to3^{+}}\frac{7x}{x - 3}=\infty$.

Step3: Analyze two - sided limit as $x\to3$

Since $\lim_{x\to3^{-}}f(x)=-\infty$ and $\lim_{x\to3^{+}}f(x)=\infty$, the two - sided limit $\lim_{x\to3}f(x)$ does not exist.

Answer:

a) $-\infty$
b) $\infty$
c) B. The limit does not exist and is neither $-\infty$ nor $\infty$.