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 :
$$=\lim_{x\to 3} \frac {\cos (\pi x)+1}{(x+5)^{\frac {1}{3}} -2}
=\lim_{x\to 3} \frac {1-(\pi x)/2! +1}{(5)^{1/3} + \dfrac {x}{3\cdot 5^{2/3}} -\dfrac {x^2}{45\cdot 5^{2/3}} -2}.$$

By definition of derivative of a  function at $x=3$ we have, 
$$\text{since $\cos3\pi =-1\implies$ }\lim_{x\to 3} \frac {\cos (\pi x)-(-1)}{x-3} =\lim_{x\to 3} -\pi\sin (\pi x) =0$$
and 
$$\lim_{x\to 3} \frac {(x+5)^{\frac {1}{3}} -2}{x-3} =\lim_{x\to 3} \frac {1}{3}(x+5)^{\frac {-2}{3}} =\frac {1}{3} 8^{-2/3} =\frac {1}{12}.$$
Therefore, $$\lim_{x\to 3} \frac {\cos (\pi x)+1}{(x+5)^{\frac {1}{3}} -2}=\lim_{x\to 3} \frac {\cos (\pi x)+1}{\color{red}{x-3}}\frac {\color{red}{x-3}}{(x+5)^{\frac {1}{3}} -2}=0$$
