<s>[INST] Calculate $\lim\limits_{x\to 3} \frac {\cos (\pi x)+1}{(x+5)^{\frac {1}{3}} -2}$ 
Calculate 
  $$\lim_{x\to 3} \frac {\cos (\pi x)+1}{(x+5)^{\frac {1}{3}} -2}.$$

My Attempt :
$$=\lim_{x\to 3} \frac {\cos (\pi x)+1}{(x+5)^{\frac {1}{3}} -2}
=\lim_{x\to 3} \frac {1-(\pi x)/2! +1}{(5)^{1/3} + \dfrac {x}{3\cdot 5^{2/3}} -\dfrac {x^2}{45\cdot 5^{2/3}} -2}.$$
 [/INST] With substitution $x=u+3$ we have
\begin{align}
\lim_{x\to 3} \dfrac {\cos (\pi x)+1}{(x+5)^{\frac {1}{3}} -2}
&= \lim_{u\to 0}\dfrac{1-\cos\pi u}{\sqrt[3]{u+8}-2} \\
&= \lim_{u\to 0}\dfrac{2\sin^2\frac12\pi u}{u^2}\dfrac{u^2}{\sqrt[3]{u+8}-2}\dfrac{\sqrt[3]{u+8}^2+\sqrt[3]{u+8}+4}{\sqrt[3]{u+8}^2+\sqrt[3]{u+8}+4} \\
&= \dfrac{\pi^2}{2}\lim_{u\to 0}\dfrac{u^2}{u}\lim_{u\to 0}\dfrac{\sqrt[3]{u+8}^2+\sqrt[3]{u+8}+4}{1} \\
&= \dfrac{\pi^2}{2}\times 0 \times 12 \\
&= 0
\end{align}
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