<s>[INST] Evaluating $\lim _{x\to \infty }\tan\left(x^2\sin\frac{\pi }{4x^2}\right)$ without L'hopital's rule How do I solve this without using L'hopital's rule?
$$\lim _{x\to \infty }\left(\tan\left(x^2\sin\left(\frac{\pi }{4x^2}\right)\right)\right)$$
Problem
EDIT:
My progress so far:
$$\lim _{x\to \infty }\left(\tan\left(x^2\sin\left(\frac{\pi }{4x^2}\right)\right)\right)$$
$$=\lim _{x\to \infty }\left(\tan\left(\frac{\sin\left(\frac{\pi }{4x^2}\right)}{\frac{1}{x^2}}\right)\right)$$
$$=\lim _{x\to \infty }\left(\tan\left(\frac{\pi}{4}*\frac{\sin\left(\frac{\pi }{4x^2}\right)}{\frac{\pi}{4x^2}}\right)\right)$$
Now I know $\frac{\sin\left(x\right)}{x} = 1$ but I don't know how to take the above expression to that form.
 [/INST] You've already taken care of one important bit, namely rewriting
$$\tan\left(x^2\sin\frac{\pi}{4x^2}\right)$$
as
$$\tan\left(\frac{\pi}{4}\cdot\frac{\sin\frac{\pi}{4x^2}}{\frac{\pi}{4x^2}}\right)$$
As you noted, $\lim_{x\to 0}\frac{\sin(x)}{x}=1$, so we should make use of this somehow. We can achieve this goal by making use of another important observation, namely that
$$\lim_{x\to\infty}\frac{\pi}{4x^2}=\frac{\pi}{4}\lim_{x\to\infty}\frac{1}{x^2}=\frac{\pi}{4}\cdot 0=0$$
Knowing this, I hope it makes sense that we should have
$$\lim_{x\to\infty}\frac{\sin\left(\frac{\pi}{4x^2}\right)}{\frac{\pi}{4x^2}}=1$$
After all, as $x$ gets bigger and bigger, $\frac{\pi}{4x^2}$ gets closer and closer to $0$, and $\sin(y)/y$ gets closer and closer to $1$ as its argument gets smaller and smaller, namely $y=\frac{\pi}{4x^2}$.
Alright, so we know that
$$\lim_{x\to\infty}\frac{\sin\left(\frac{\pi}{4x^2}\right)}{\frac{\pi}{4x^2}}=1$$
and thus, by the constant multiple rule for limits
$$\lim_{x\to\infty}\left(\frac{\pi}{4}\cdot\frac{\sin\left(\frac{\pi}{4x^2}\right)}{\frac{\pi}{4x^2}}\right)=\frac{\pi}{4}$$
But the tangent function $\tan$ is continuous near $\pi/4$, so
$$\lim_{x\to\infty}\tan\left(\frac{\pi}{4}\cdot\frac{\sin\left(\frac{\pi}{4x^2}\right)}{\frac{\pi}{4x^2}}\right)=\tan\left(\lim_{x\to\infty}\left(\frac{\pi}{4}\cdot\frac{\sin\left(\frac{\pi}{4x^2}\right)}{\frac{\pi}{4x^2}}\right)\right)=\tan\left(\frac{\pi}{4}\right)=1$$
We conclude that the original limit is
$$\lim_{x\to\infty}\tan\left(x^2\sin\frac{\pi}{4x^2}\right)=1$$
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