find the limits in a), b), and c) below for the function (f(x)=\frac{5x}{x - 6}). use (-infty) and (infty)…

find the limits in a), b), and c) below for the function (f(x)=\frac{5x}{x - 6}). use (-infty) and (infty) when appropriate.\nb) select the correct choice below and fill in any answer boxes in your choice.\nthe limit does not exist and is neither (-infty) nor (infty).\n(simplify your answer.)\na. (lim_{x\rightarrow6^{+}}f(x)=infty)\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\rightarrow6^{-}}f(x)=)\n(simplify your answer.)\nb. the limit does not exist and is neither (-infty) nor (infty).

find the limits in a), b), and c) below for the function (f(x)=\frac{5x}{x - 6}). use (-infty) and (infty) when appropriate.\nb) select the correct choice below and fill in any answer boxes in your choice.\nthe limit does not exist and is neither (-infty) nor (infty).\n(simplify your answer.)\na. (lim_{x\rightarrow6^{+}}f(x)=infty)\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\rightarrow6^{-}}f(x)=)\n(simplify your answer.)\nb. the limit does not exist and is neither (-infty) nor (infty).

Answer

  1. First, consider the function (f(x)=\frac{5x}{x - 6}).
    • a) Find (\lim_{x\rightarrow6^{+}}f(x)):
      • As (x\rightarrow6^{+}), we can analyze the behavior of the function. Let (x = 6 + h), where (h>0) and (h\rightarrow0).
      • Then (f(x)=\frac{5(6 + h)}{(6 + h)-6}=\frac{30 + 5h}{h}).
      • As (h\rightarrow0^{+}), (\frac{30 + 5h}{h}=\frac{30}{h}+5). Since (\frac{30}{h}\rightarrow+\infty) as (h\rightarrow0^{+}), (\lim_{x\rightarrow6^{+}}f(x)=\infty).
    • b) Find (\lim_{x\rightarrow6^{-}}f(x)):
      • Let (x = 6 - h), where (h>0) and (h\rightarrow0).
      • Then (f(x)=\frac{5(6 - h)}{(6 - h)-6}=\frac{30 - 5h}{-h}=-\frac{30}{h}+5).
      • As (h\rightarrow0^{+}), (-\frac{30}{h}\rightarrow-\infty), so (\lim_{x\rightarrow6^{-}}f(x)=-\infty).
    • c) Find (\lim_{x\rightarrow6}f(x)):
      • Since (\lim_{x\rightarrow6^{+}}f(x)=\infty) and (\lim_{x\rightarrow6^{-}}f(x)=-\infty), the two - sided limit (\lim_{x\rightarrow6}f(x)) does not exist and is neither (-\infty) nor (\infty).

Explanation:

Step1: Analyze right - hand limit

Let (x = 6 + h) ((h\rightarrow0^{+})), (f(x)=\frac{5(6 + h)}{h}=\frac{30}{h}+5\rightarrow\infty).

Step2: Analyze left - hand limit

Let (x = 6 - h) ((h\rightarrow0^{+})), (f(x)=\frac{5(6 - h)}{-h}=-\frac{30}{h}+5\rightarrow-\infty).

Step3: Determine two - sided limit

Since left - hand and right - hand limits are not equal, (\lim_{x\rightarrow6}f(x)) does not exist and is neither (-\infty) nor (\infty).

Answer:

a) A. (\lim_{x\rightarrow6^{+}}f(x)=\infty) b) A. (\lim_{x\rightarrow6^{-}}f(x)=-\infty) c) B. The limit does not exist and is neither (-\infty) nor (\infty)