<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] $$\lim_{x\to 3} \dfrac {\cos (\pi x)+1}{(x+5)^{1/3} -2}=\frac{\pi^2}{2}\lim_{x\rightarrow3}\left(\frac{\sin^2\frac{\pi(x-3)}{2}}{\frac{\pi^2(x-3)^2}{4}}\cdot\frac{(x-3)^2}{\sqrt[3]{x+5}-2}\right)=$$
$$=\frac{\pi^2}{2}\lim_{x\rightarrow3}(x-3)\left(\sqrt[3]{(x+5)^2}+2\sqrt[3]{x+5}+4\right)=0.$$
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