<s>[INST] Find the limit of $ \lim_{x \to 7} \frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2} $ I need to evaluate the limit without using L'Hopital's rule.
$$\Large
\lim_{x \to 7} \frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}
$$
 [/INST] Sorry to add a second answer, dont upvote if you dont like it, but this answer is different from my first and is essentially the same as some of the other answers, but I wanted to point a different formulation that is a little to long for a comment.
One can use the limit
$$\lim\limits_{y \to 0} \frac{(y+k)^{\alpha}-k^{\alpha}}{y}=\alpha k^{\alpha -1}$$
then with the substitution, $y=x-7$ we can write the expression as
$$\frac{\left( \frac{\sqrt{y+9}-3}{y} \right) -\left( \frac{\sqrt[3]{y+27}-3}{y} \right)}{\frac{\sqrt[4]{y+16}-2}{y} }$$
and this gives, 
$$\frac{\frac{1}{2}9^{-\frac{1}{2}}-\frac{1}{3}27^{-\frac{2}{3}} }{\frac{1}{4}16^{-\frac{3}{4}}}=\frac{112}{27}$$
</s>