# Find the following limits or state that they do not exist. lim_{xrightarrow4}frac{31x-42cdotsqrt{x+5}}{3-sqrt{x+5}}

Find the following limits or state that they do not exist. $\underset{x\to 4}{lim}\frac{31x-42\cdot \sqrt{x+5}}{3-\sqrt{x+5}}$
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berggansS
Consider the provide expression,
$\underset{x\to 4}{lim}\frac{31x-42\cdot \sqrt{x+5}}{3-\sqrt{x+5}}$
Now, find the limit of the provided expression.
$\underset{x\to 4}{lim}\frac{31x-42\cdot \sqrt{x+5}}{3-\sqrt{x+5}}$
Its is not determinant form because the expression from is $\frac{0}{0}$
Apply the L Hospital rule,
$\underset{x\to 4}{lim}\frac{31x-42\cdot \sqrt{x+5}}{\left(3-\sqrt{x+5}}=\underset{x\to 4}{lim}\frac{31-42}{\left(2\sqrt{x+5}\right)}\left(3-\frac{1}{\sqrt{x+5}}\right)$
So, it is a determinant form
$\underset{x\to 4}{lim}\frac{31x-42\cdot \sqrt{x+5}}{3-\sqrt{x+5}}=\underset{x\to 4}{lim}\frac{31-\frac{42}{2\cdot \sqrt{x+5}}}{3-\frac{1}{\sqrt{x+5}}}$
$=\frac{31-\frac{42}{6}}{3-\frac{1}{3}}$
$=\frac{24\cdot 3}{8}$
$=9$
Hence, the limit is 9
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