Evaluate the following limits: _ x 4 [x^2+x-20 3 x+4-4 ] - Vidyadip

Question

Evaluate the following limits: $\lim _{x \rightarrow 4}\left[\frac{x^2+x-20}{\sqrt{3 x+4}-4}\right]$

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Answer

$\lim _{x \rightarrow 4}\left[\frac{x^2+x-20}{\sqrt{3 x+4}-4}\right]$
$=\lim _{x \rightarrow 4}\left[\frac{(x+5)(x-4)}{\sqrt{3 x+4}-4} \times \frac{\sqrt{3 x+4}+4}{\sqrt{3 x+4}+4}\right]$
$=\lim _{x \rightarrow 4} \frac{(x+5)(x-4)(\sqrt{3 x+4}+4)}{(3 x+4)-16}$
$=\lim _{x \rightarrow 4} \frac{(x+5)(x-4)(\sqrt{3 x+4}+4)}{3 x-12}$
$=\lim _{x \rightarrow 4} \frac{(x+5)(x-4)(\sqrt{3 x+4}+4)}{3(x-4)}$
$=\lim _{x \rightarrow 4} \frac{(x+5)(\sqrt{3 x+4}+4)}{3} \quad \cdots\left[\begin{array}{l}
\because x \rightarrow 4, x \neq 4 \\ \therefore x-4 \neq 0 \end{array}\right]$
$=\frac{(4+5)(\sqrt{3(4)+4}+4)}{3}$
$=\frac{9(4+4)}{3}$
$=24...[$ By rationalization$]$

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