Calculate the limit without l'Hopital rule I have the following limit:
$$
\lim_{x\to 7}\dfrac{x^2-4x-21}{x-4-\sqrt{x+2}}
$$
I could easily calculate the limit = 12 using the l'Hopital  rule.
Could you please suggest any other ways to solve this limit without using the l'Hopital rule?
Thank you

The numerator can be factored as $(x-7)(x+3)$. Now consider
$$
\lim_{x\to7}\frac{x-4-\sqrt{x+2}}{x-7}=\lim_{x\to7}\frac{x-7-(\sqrt{x+2}-3)}{x-7}=
1-\lim_{x\to7}\frac{x+2-9}{(x-7)(\sqrt{x+2}+3)}=1-\frac{1}{6}=\frac{5}{6}
$$
So your limit is
$$
\lim_{x\to7}\frac{x-7}{x-4-\sqrt{x+2}}(x+3)=\frac{6}{5}\cdot10=12
$$
