let (f(x)=\frac{x - 6}{x^{2}-6x}). find the indicated quantities, if they exist.\n(a) (lim_{x\rightarrow0}f(x…

let (f(x)=\frac{x - 6}{x^{2}-6x}). find the indicated quantities, if they exist.\n(a) (lim_{x\rightarrow0}f(x)) (b) (lim_{x\rightarrow6}f(x)) (c) (lim_{x\rightarrow36}f(x)\n(a) select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. (lim_{x\rightarrow0}\frac{x - 6}{x^{2}-6x}=square) (type an integer or a simplified fraction.)\nb. the limit does not exist

let (f(x)=\frac{x - 6}{x^{2}-6x}). find the indicated quantities, if they exist.\n(a) (lim_{x\rightarrow0}f(x)) (b) (lim_{x\rightarrow6}f(x)) (c) (lim_{x\rightarrow36}f(x)\n(a) select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. (lim_{x\rightarrow0}\frac{x - 6}{x^{2}-6x}=square) (type an integer or a simplified fraction.)\nb. the limit does not exist

Answer

Explanation:

Step1: Simplify the function

First, factor the denominator of (f(x)=\frac{x - 6}{x^{2}-6x}). Since (x^{2}-6x=x(x - 6)), then (f(x)=\frac{x - 6}{x(x - 6)}=\frac{1}{x}), for (x\neq6).

Step2: Calculate (\lim_{x\rightarrow0}f(x))

Substitute (x = 0) into (\frac{1}{x}). As (x\rightarrow0), (\frac{1}{x}) approaches either positive or negative infinity depending on the side - approach. So (\lim_{x\rightarrow0}\frac{x - 6}{x^{2}-6x}) does not exist.

Step3: Calculate (\lim_{x\rightarrow6}f(x))

Substitute (x = 6) into (\frac{1}{x}). (\lim_{x\rightarrow6}\frac{1}{x}=\frac{1}{6}).

Step4: Calculate (\lim_{x\rightarrow36}f(x))

Substitute (x = 36) into (\frac{1}{x}). (\lim_{x\rightarrow36}\frac{1}{x}=\frac{1}{36}).

Answer:

(A) B. The limit does not exist (B) (\frac{1}{6}) (C) (\frac{1}{36})