Trigonometric Limit: $\lim_{x\to 0} \frac{\sqrt {\cos x} - \sqrt[3] {\cos x}}{\sin^2x}$

Question

Compute the following limit:

$$\lim_{x\to 0} \frac{\sqrt {\cos x} - \sqrt[3] {\cos x}}{\sin^2x}$$

How would I go about solving this, I can't used l´Hôpital

Answer 1

Hint: $\lim_{x\to 0} \frac{1-\cos x}{x^2}=\frac12$, i.e. $\cos x\sim 1-x^2/2$ as $x\to 0$. Hence $$ \frac{\cos^{1/2}x - \cos^{1/3}x}{\sin^2 x} \sim \frac{(1-x^2/2)^{1/2}-(1-x^2/2)^{1/3}}{x^2}$$ and use $(1+x)^a\sim 1+ax$ for $x\to 0$...

Answer 2

Writing $c$ and $s$ for $\cos x$ and $\sin x$ respectively,

$$\frac{c^{1/2} - c^{1/3}}{s^2} = \frac{c^{1/3}}{s^2} (c^{1/6} - 1) = \frac{c -1}{s^2} \frac{c^{1/3}}{c^{5/6} + c^{4/6} + c^{3/6} + c^{2/6} + c^{1/6} +1} \ \ \ --(*) $$

as $(c^{1/6} - 1)(c^{5/6} + c^{4/6} + c^{3/6} + c^{2/6} + c^{1/6} +1) = c - 1$

Now $$c - 1 = \cos x - 1 = -2\sin^2(x/2) \ \ \hbox{ and} \ \ s^2 = \sin^2 x = 4\sin^2(x/2)\cos^2(x/2)$$ Hence the first fraction of (*)

$$ \frac{c -1}{s^2} = -\frac{1}{2}\cos^2(x/2) \longrightarrow -\frac{1}{2}, \ \ \hbox{ as } x \rightarrow 0$$

The second fraction $\displaystyle \frac{c^{1/3}}{c^{5/6} + c^{4/6} + c^{3/6} + c^{2/6} + c^{1/6} +1} \ \longrightarrow \frac{1}{6}, \ \ \hbox{ as } x \rightarrow 0$

Hence $$\lim_{x\rightarrow 0} \frac{c^{1/2} - c^{1/3}}{s^2} = -\frac{1}{2} . \frac{1}{6} = -\frac{1}{12}$$

Answer 3

Set $\sqrt[6]{\cos\left(x\right)}↦\ t$ then: $$\lim_{x\to 0} \frac{\sqrt {\cos x} - \sqrt[3] {\cos x}}{\sin^2x}$$$$=\lim_{t\to 1}\frac{t^{3}-t^{2}}{1-t^{12}}$$$$=-\lim_{t\to 1}\frac{t^{2}\left(t-1\right)}{\left(t-1\right)\left(\sum_{n=0}^{11}t^{n}\right)}=-\frac{1}{12}$$