<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] $$\lim_{x\rightarrow0}\frac{1 + \tan x + \sin x - \cos x}{\sin^2 x + x^3}=\lim_{x\rightarrow0}\frac{2\sin^2\frac{x}{2} +2\sin\frac{x}{2}\cos\frac{x}{2}\left(\frac{1}{\cos{x}}+1\right) }{\sin^2 x + x^3}=$$
$$=\lim_{x\rightarrow0}\left(\frac{\sin\frac{x}{2}}{\frac{x}{2}}\cdot\frac{\sin\frac{x}{2} +\cos\frac{x}{2}\left(\frac{1}{\cos{x}}+1\right) }{\frac{\sin x}{x}\cdot\sin{x} + x^2}\right)=\infty$$
because $\frac{\sin{x}}{x}\rightarrow1$.
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