<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] \begin{align}
\dfrac{1-\cos x+\sin x+\tan x}{\sin^2x+x^3}
&= \dfrac{2\sin^2\frac{x}{2}+\sin x\dfrac{1+\cos x}{\cos x}}{\sin^2x+x^3} \\
&= \dfrac{2\dfrac{\sin^2\frac{x}{2}}{x^2}+\dfrac{\sin x}{x}\dfrac{1+\cos x}{x\cos x}}{\dfrac{\sin^2x}{x^2}+x} \\
&\to\dfrac{\dfrac12+1\times\infty}{1+0}=\infty
\end{align}
as $x\to0$.
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