TITLE: Calculate $\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}$
QUESTION [12 upvotes]: Please help me calculate this:
$$\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}$$
Here I've tried multiplying by $\sqrt[4]{x+9}+2$ and few other method. 
Thanks in advance for solution / hints using simple methods.
Edit
Please don't use l'Hosplital rule. We are before derivatives, don't know how to use it correctly yet. Thanks!

REPLY [6 votes]: $$\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}$$
$$=\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{{(x+9)^\frac{1}{4}}-(16)^\frac{1}{4}}.\frac{x-7}{x-7}$$
$$=\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{x-7}.\underset{x\rightarrow7}{\lim}\frac{x-7}{{(x+9)^\frac{1}{4}}-(16)^\frac{1}{4}}$$
$$=\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}-3+3}{x-7}.\underset{x\rightarrow7}{\lim}\frac{x+9-16}{{(x+9)^\frac{1}{4}}-(16)^\frac{1}{4}}$$
$$=[\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt9}{x-7}-\underset{x\rightarrow7}{\lim}\frac{\sqrt[3]{x+20}-\sqrt[3]27}{x-7}].\underset{x\rightarrow7}{\lim}\frac{(x+9)-16}{{(x+9)^\frac{1}{4}}-(16)^\frac{1}{4}}$$
$$= \frac{112}{27}$$