Compute $\lim_{x\to 0}\dfrac{\sqrt[m]{\cos x}-\sqrt[n]{\cos x}}{x^2}$ For two positive integers $m$ and $n$, compute
$$\lim_{x\to 0}\dfrac{\sqrt[m]{\cos x}-\sqrt[n]{\cos x}}{x^2}$$  
Without loss of generality I consider $m>n$ and multiply the numerator with its conjugate. But what next? Cannot proceed further! Help please!

Since $x$ is going to zero, expand $\cos x$ as a Taylor series (one term would be sufficient) and use the fact that, for small values of $y$, $(1-y)^a$ is close to $(1-a y)$ (this is also coming from a Taylor series). So, you will easily establish that
$\cos^{1/m}(x) = 1- \dfrac{x^2}{2m}$
Doing the same for $n$, you end with $\dfrac{m - n}{2mn}$
For sure this only applies if $m$ is not equal to $n$.
