# Use L'Hopital's Rule to work the limits. a) \lim _{x - 2} {x^{2}-2 x}/{x^{2}-x-2} b) \lim _{x -...

## Question:

Use L'Hopital's Rule to work the limits.

a){eq}\lim _{x \rightarrow 2} \frac{x^{2}-2 x}{x^{2}-x-2} {/eq}

b) {eq}\lim _{x \rightarrow \infty} \frac{1-2 x^{2}}{x^{2}+1} {/eq}

c){eq}\lim _{x \rightarrow \infty} x \ln \left(1+\frac{1}{\sqrt{x}}\right) {/eq}

## Limits:

In order to find the value of the given limits, we will make use of the L' Hospital's rule. This rule is applied to make the given function solvable, which is otherwise not. In this rule, we take the derivative of the numerator and the denominator terms separately.

## Answer and Explanation:

a) {eq}\lim _{x \rightarrow 2} \frac{x^{2}-2 x}{x^{2}-x-2} {/eq}

Applying L'Hospital's rule, we get:

{eq}\begin{align*} \ & \ \lim _{x...

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