TITLE: Limit of composite trigonometric function
QUESTION [1 upvotes]: I am trying to find an easy way to compute the limit as $x \to 0$ of 
$$f(x) = \frac{\sqrt{1+\tan(x)} - \sqrt{1+\sin(x)}}{x^3}$$
from first principles (i.e. without using l'Hôspital's rule).
I have gone as far as boiling down the problem to computing the limit as $x \to 0$ of
$$\frac{1 - \cos(x)}{x^2}$$
I thought about using the Small Angle Approximation for cosine, which indeed gives the right answer but doesn't seem to be a very formal.
Any hint?
Also, my working was fairly long so if you have a straightforward way to compute the limit of $f(x)$ I would love to hear it :)

REPLY [1 votes]: Try using double-angle formulae.
$1 - \cos x = 1 - \left ( 1- 2 \sin^2 \left ( \frac{\theta}{2} \right ) \right ) = 2 \sin^2 \left ( \frac{\theta}{2} \right )$
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