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) select the correct choice below and fill in any answer boxes in your choice.
a. \lim_{x\to3^{-}}f(x)=-\infty
(simplify your answer.)
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)=
(simplify your answer.)
b. the limit does not exist and is neither -\infty nor \infty.

Explanation:

Step1: Analyze left - hand limit

As \(x\to3^{-}\), the numerator \(7x\to 7\times3 = 21\) (a positive number), and the denominator \(x - 3\to0^{-}\) (a very small negative number). So \(\lim_{x\to3^{-}}\frac{7x}{x - 3}=-\infty\).

Step2: Analyze right - hand limit

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

Answer:

a) A. \(\lim_{x\to3^{-}}f(x)=-\infty\)
b) A. \(\lim_{x\to3^{+}}f(x)=\infty\)