Mathematical Analysis I - 1-5-1.(14)

$\begin{aligned} \underset{x\to 2}{\mathop{\lim }}\,\dfrac{\sqrt{x+2}-2}{\sqrt{x+7}-3} & =\underset{x\to 2}{\mathop{\lim }}\,\dfrac{\left( \sqrt{x+2}-2 \right)\cdot \left( {\color{#3D99F6}{\sqrt{x+2}+2}} \right)\cdot \left( {\color{#D61F06}{\sqrt{x+7}+3}} \right)}{\left( \sqrt{x+7}-3 \right)\cdot \left( {\color{#D61F06}{\sqrt{x+7}+3}} \right)\cdot \left( {\color{#3D99F6}{\sqrt{x+2}+2}} \right)} \\ & =\underset{x\to 2}{\mathop{\lim }}\,\dfrac{\cancel{\left( x-2 \right)}\cdot \left( \sqrt{x+7}+3 \right)}{\cancel{\left( x-2 \right)}\cdot \left( \sqrt{x+2}+2 \right)} \\ & =\underset{x\to 2}{\mathop{\lim }}\,\dfrac{\sqrt{x+7}+3}{\sqrt{x+2}+2} \\ & =\dfrac{3+3}{2+2} \\ & =\boxed{1.50}. \\ \end{aligned}$

$\begin{aligned} \lim_{x \to 2}\frac {\sqrt{x+2}-2}{\sqrt{x+7}-3} & = \lim_{x \to 2} \frac {(\sqrt{x+2}-2)(\sqrt{x+2}+2)}{(\sqrt{x+7}-3)(\sqrt{x+2}+2)} \\ & = \lim_{x \to 2}\frac 1{\sqrt{x+2}+2} \cdot \lim_{x \to 2}\frac {x-2}{\sqrt{x+7}-3} \\ & = \frac 14 \lim_{x \to 2}\frac {(x-2)(\sqrt{x+7}+3)}{(\sqrt{x+7}-3)(\sqrt{x+7}+3)} \\ & = \frac 14 \lim_{x \to 2}\frac {(x-2)(\sqrt{x+7}+3)}{x-2} \\ & = \frac 14 \lim_{x \to 2} (\sqrt{x+7}+3) \\ & = \frac 14 \cdot 6 = \frac 32 = \boxed {1.5} \end{aligned}$

Let $x=2+y$ where $y$ is so small that terms containing powers of $y$ greater than $1$ can be neglected. Then the given limit is

$\displaystyle \lim_{y\to 0} \dfrac 23 \times \dfrac {1+\frac y8 -1}{1+\frac{y}{18}-1}$

$=\dfrac 23\times \dfrac {18}{8}=\dfrac 32=\boxed {1.5}$ .

There may be a more elegant method of solution.