<s>[INST] 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?
 [/INST] Let $f(x)=\sqrt{x+2}-\sqrt[3]{x+20}$ and $g(x)=\sqrt[4]{x+9}-2$, then 
\begin{align*}
\lim_{x\rightarrow 7}\dfrac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}&=\lim_{x\rightarrow 7}\dfrac{f(x)-f(7)}{x-7}\lim_{x\rightarrow 7}\dfrac{x-7}{g(x)-g(7)}\\
&=f'(7)\times\dfrac{1}{g'(7)}.
\end{align*}
Alternatively, if we let 
\begin{align*}
\dfrac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}&=\dfrac{\dfrac{u^{1/2}-v^{1/2}}{u^{1/3}+u^{1/6}v^{1/6}+v^{1/3}}}{\dfrac{x-7}{(x+9)^{1/4}+2}\dfrac{1}{(x+9)^{1/2}+4}},
\end{align*}
where $u=(x+2)^{3}$, $v=(x+20)^{2}$, now $u^{1/2}-v^{1/2}=\dfrac{u-v}{u^{1/2}+v^{1/2}}$, where $u-v=(x-7)(x^{2}+12x+56)$.
</s>