<s>[INST] Limit of $f(x)$ when $x$ goes to zero Let $f(x) = \frac{1 + \tan x + \sin x - \cos x}{\sin^2 x + x^3}$ . Find value of $\lim_{x \to 0} f(x)$ if it exists . I can solve it using L'Hospital's Rule and  Taylor series but I'm looking for another way suing trigonometric identities .
 [/INST] As $x\to 0$ we have 
$\tan x\sim x;\;\sin x \sim x;\;1-\cos x\sim \dfrac{x^2}{2}$
So the limit becomes
$$\lim_{x\to 0} \frac{1 + \tan x + \sin x - \cos x}{\sin^2 x + x^3}=\lim_{x\to 0}\frac{x+x+\frac{x^2}{2}}{x^2+x^3}=\lim_{x\to 0}\frac{4x+x^2}{2(x^2+x^3)}=\lim_{x\to 0}\frac{x(4+x)}{2x^2(1+x)}=$$
$$=\lim_{x\to 0}\frac{4+x}{2x(1+x)}=\infty$$
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