Question

find the limits in a), b), and c) below for the function (f(x)=\frac{5x}{x - 6}). use (-infty) and (infty) when appropriate.
b) select the correct choice below and fill in any answer boxes in your choice.
the limit does not exist and is neither (-infty) nor (infty).
(simplify your answer.)
a. (lim_{x
ightarrow6^{+}}f(x)=infty)
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
ightarrow6^{-}}f(x)=)
(simplify your answer.)
b. the limit does not exist and is neither (-infty) nor (infty).

Explanation:

1. First, consider the function \(f(x)=\frac{5x}{x - 6}\).

• **a) Find \(\lim_{x

ightarrow6^{+}}f(x)\)**:

• As \(x

ightarrow6^{+}\), we can analyze the behavior of the function. Let \(x = 6 + h\), where \(h>0\) and \(h
ightarrow0\).

• Then \(f(x)=\frac{5(6 + h)}{(6 + h)-6}=\frac{30 + 5h}{h}\).
• As \(h

ightarrow0^{+}\), \(\frac{30 + 5h}{h}=\frac{30}{h}+5\). Since \(\frac{30}{h}
ightarrow+\infty\) as \(h
ightarrow0^{+}\), \(\lim_{x
ightarrow6^{+}}f(x)=\infty\).

• **b) Find \(\lim_{x

ightarrow6^{-}}f(x)\)**:

• Let \(x = 6 - h\), where \(h>0\) and \(h

ightarrow0\).

• Then \(f(x)=\frac{5(6 - h)}{(6 - h)-6}=\frac{30 - 5h}{-h}=-\frac{30}{h}+5\).
• As \(h

ightarrow0^{+}\), \(-\frac{30}{h}
ightarrow-\infty\), so \(\lim_{x
ightarrow6^{-}}f(x)=-\infty\).

• **c) Find \(\lim_{x

ightarrow6}f(x)\)**:

• Since \(\lim_{x

ightarrow6^{+}}f(x)=\infty\) and \(\lim_{x
ightarrow6^{-}}f(x)=-\infty\), the two - sided limit \(\lim_{x
ightarrow6}f(x)\) does not exist and is neither \(-\infty\) nor \(\infty\).

Step1: Analyze right - hand limit

Let \(x = 6 + h\) (\(h
ightarrow0^{+}\)), \(f(x)=\frac{5(6 + h)}{h}=\frac{30}{h}+5
ightarrow\infty\).

Step2: Analyze left - hand limit

Let \(x = 6 - h\) (\(h
ightarrow0^{+}\)), \(f(x)=\frac{5(6 - h)}{-h}=-\frac{30}{h}+5
ightarrow-\infty\).

Step3: Determine two - sided limit

Since left - hand and right - hand limits are not equal, \(\lim_{x
ightarrow6}f(x)\) does not exist and is neither \(-\infty\) nor \(\infty\).

Answer:

a) A. \(\lim_{x
ightarrow6^{+}}f(x)=\infty\)
b) A. \(\lim_{x
ightarrow6^{-}}f(x)=-\infty\)
c) B. The limit does not exist and is neither \(-\infty\) nor \(\infty\)