How can I calculate the limit $\lim\limits_{x \to 7} \frac{\sqrt{x+2} - \sqrt[3]{x+20}}{\sqrt[4]{x+9} - 2}$ without L'Hospital's rule? I have a problem with calculation of the limit:

$$\lim\limits_{x \to 7} \frac{\sqrt{x+2} - \sqrt[3]{x+20}}{\sqrt[4]{x+9} - 2}$$

Is there a way to calculate it? How can I do it?

Hint:
If $\sqrt{x+2}=a,\sqrt[3]{x+20}=b,\sqrt[4]{x+9}=c$
LCM$(2,3)=6$
$a^6-b^6=(a-b)(\cdots)$
Similarly $c^4-2^4=(c-2)\cdots$
