evaluate the limit, if it exists. (if an answer does not exist, enter dne.)\nlim(x→36) (6 - √x)/(36x - x²)

Explanation:

Step1: Factor the denominator

Factor out $x$ from $36x - x^{2}$: $36x - x^{2}=x(36 - x)$. The limit becomes $\lim_{x\rightarrow36}\frac{6-\sqrt{x}}{x(36 - x)}$.

Step2: Rationalize the numerator

Multiply the numerator and denominator by the conjugate of the numerator $6 + \sqrt{x}$. [ \begin{align*} &\lim_{x\rightarrow36}\frac{(6-\sqrt{x})(6 + \sqrt{x})}{x(36 - x)(6+\sqrt{x})}\ =&\lim_{x\rightarrow36}\frac{36 - x}{x(36 - x)(6+\sqrt{x})} \end{align*} ]

Step3: Simplify the expression

Cancel out the common factor $(36 - x)$ (since $x\neq36$ when taking the limit), we get $\lim_{x\rightarrow36}\frac{1}{x(6+\sqrt{x})}$.

Step4: Substitute $x = 36$

Substitute $x = 36$ into $\frac{1}{x(6+\sqrt{x})}$, we have $\frac{1}{36\times(6 + \sqrt{36})}=\frac{1}{36\times(6 + 6)}=\frac{1}{36\times12}=\frac{1}{432}$.

Answer:

$\frac{1}{432}$