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

Question
Limits and continuity
Find the following limits or state that they do not exist. $$\lim_{x\rightarrow4}\frac{31x-42\cdot\sqrt{x+5}}{3-\sqrt{x+5}}$$

2021-02-26
Consider the provide expression,
$$\lim_{x\rightarrow4}\frac{31x-42\cdot\sqrt{x+5}}{3-\sqrt{x+5}}$$
Now, find the limit of the provided expression.
$$\lim_{x\rightarrow4}\frac{31x-42\cdot\sqrt{x+5}}{3-\sqrt{x+5}}$$
Its is not determinant form because the expression from is $$\frac{0}{0}$$
Apply the L Hospital rule,
$$\lim_{x\rightarrow4}\frac{31x-42\cdot\sqrt{x+5}}{(3-\sqrt{x+5}}=\lim_{x\rightarrow4}\frac{31-42}{(2\sqrt{x+5})}(3-\frac{1}{\sqrt{x+5}})$$
So, it is a determinant form
$$\lim_{x\rightarrow4}\frac{31x-42\cdot\sqrt{x+5}}{3-\sqrt{x+5}}=\lim_{x\rightarrow4}\frac{31-\frac{42}{2\cdot\sqrt{x+5}}}{3-\frac{1}{\sqrt{x+5}}}$$
$$=\frac{31-\frac{42}{6}}{3-\frac{1}{3}}$$
$$=\frac{24\cdot3}{8}$$
$$=9$$
Hence, the limit is 9

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