<s>[INST] Evaluate $ \lim_{x\to 0} \frac{\tan(4x)}{\sin(7x)}$ Evaluate $$ \lim_{x\to 0} \  \frac{\tan(4x)}{\sin(7x)}$$
I am stuck after I convert tan(4x) into sin (4x) / cosine(4x)  
 [/INST] Recall that $\lim_{x\to0} \sin(x)/x = 1$ and $\lim_{x\to 0} \tan(x)/x = 1$.
Then $$\lim_{x\to 0} \frac{\tan(4x)}{\sin(7x)} = \lim_{x\to 0} \frac{7}{7} \cdot \frac{4x}{4x} \cdot \frac{\tan(4x)}{\sin(7x)} = \lim_{x\to 0} \frac{4}{7} \frac{\tan(4x)}{4x} \cdot \frac{7x}{\sin(7x)} = \frac47 \cdot 1 \cdot 1 = \frac47.$$

Once we accept that $\lim_{x\to0} \frac{\sin(x)}{x} =1$ it immediately follows that $\lim_{x\to0} \frac{\tan(x)}{x} = 1$, since $$\lim_{x\to0} \frac{\tan(x)}{x} = \lim_{x\to0} \frac{\sin(x)}{x} \cdot \frac{1}{\cos(x)} = 1 \cdot 1 = 1$$
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