use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before…

Explanation:

Step1: Simplify the numerator of the first - limit

First, simplify $\frac{1}{x}-\frac{1}{9}=\frac{9 - x}{9x}$. Then the limit $\lim_{x\rightarrow9}\frac{\frac{1}{x}-\frac{1}{9}}{x - 9}=\lim_{x\rightarrow9}\frac{\frac{9 - x}{9x}}{x - 9}=\lim_{x\rightarrow9}\frac{9 - x}{9x(x - 9)}$. Factor out - 1 from the numerator: $\lim_{x\rightarrow9}\frac{-(x - 9)}{9x(x - 9)}$.

Step2: Cancel out the common factor

Cancel out the common factor $(x - 9)$ (since $x\neq9$ when taking the limit). We get $\lim_{x\rightarrow9}\frac{-1}{9x}$.

Step3: Evaluate the limit

Substitute $x = 9$ into $\frac{-1}{9x}$. We have $\frac{-1}{9\times9}=-\frac{1}{81}$.

Step4: Simplify the second - limit

For $\lim_{x\rightarrow4}\frac{\sqrt{x}-2}{x - 4}$, multiply the numerator and denominator by the conjugate of the numerator $\sqrt{x}+2$. We get $\lim_{x\rightarrow4}\frac{(\sqrt{x}-2)(\sqrt{x}+2)}{(x - 4)(\sqrt{x}+2)}$.

Step5: Use the difference - of - squares formula

By the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, the numerator $(\sqrt{x}-2)(\sqrt{x}+2)=x - 4$. So the limit becomes $\lim_{x\rightarrow4}\frac{x - 4}{(x - 4)(\sqrt{x}+2)}$.

Step6: Cancel out the common factor

Cancel out the common factor $(x - 4)$ (since $x\neq4$ when taking the limit). We have $\lim_{x\rightarrow4}\frac{1}{\sqrt{x}+2}$.

Step7: Evaluate the limit

Substitute $x = 4$ into $\frac{1}{\sqrt{x}+2}$. We get $\frac{1}{\sqrt{4}+2}=\frac{1}{2 + 2}=\frac{1}{4}$.

Answer:

41. $-\frac{1}{81}$
42. $\frac{1}{4}$