<s>[INST] What am I doing wrong in calculating the following limit? $$\lim_{x\to-2} \frac{x+2}{\sqrt{6+x}-2}=\lim_{x\to-2} \frac{1+2/x}{\sqrt{(6/x^2)+(1/x)}-2/x^2}$$ Dividing numerator and denominator by $x \neq0$
$$\frac{1+2/-2}{\sqrt{(6/4)+(1/-2)}-2/4}=\frac{0}{1/2}=0$$ but the limit is $4$ according to Wolfram Alpha?
 [/INST] You say you divide by $x$, but that's not what you do in the denominator; it would be:
$$\lim_{x\to-2} \frac{x+2}{\sqrt{6+x}-2}
=\lim_{x\to-2} \frac{1+\tfrac{2}{x}}{\tfrac{\sqrt{6+x}}{x}-\tfrac{2}{x}} =\lim_{x\to-2} \frac{1+\tfrac{2}{x}}{-\sqrt{\tfrac{6}{x^2}+\tfrac{1}{x}}-\tfrac{2}{x}} $$
A better approach:
$$\begin{array}{rl}
\displaystyle \lim_{x\to-2} \frac{x+2}{\sqrt{6+x}-2}
& \displaystyle = \lim_{x\to-2} \frac{\left(x+2\right)\color{blue}{\left(\sqrt{6+x}+2\right)}}{\left(\sqrt{6+x}-2\right)\color{blue}{\left(\sqrt{6+x}+2\right)}} \\[7pt]
& \displaystyle = \lim_{x\to-2} \frac{\left(x+2\right)\left(\sqrt{6+x}+2\right)}{x+2} \\[7pt]
& \displaystyle = \lim_{x\to-2} \left(\sqrt{6+x}+2\right) \\
& = 4
\end{array}$$
</s>