<s>[INST] How to find an analytic solution to $\lim_{x \to +\infty} \frac{x+\sin(x^2)}{\sqrt{x^2+1}}$ To solve this limit:
$$\lim_{x \to +\infty} \space \frac{x+\sin(x^2)}{\sqrt{x^2+1}}$$
At the beginning I didn't know how to start. Then I thought, no matter the value that $x$ takes, $sin(x^2)$ will always be between $-1$ and $1$. So for large values of $x$, $sin(x^2)$ is insignificant. One can rewrite:
$$\lim_{x \to +\infty} \space \frac{x}{\sqrt{x^2+1}}$$
And now it's easy to find the limit:
$$\lim_{x \to +\infty} \space \frac{x}{\sqrt{x^2+1}} = \lim_{x \to +\infty} \space \frac{\sqrt{x^2}}{\sqrt{x^2+1}} = \lim_{x \to +\infty} \space \sqrt{\frac{x^2}{x^2+1}} = \lim_{x \to +\infty} \space \sqrt{\frac{x^2}{x^2}}=1$$
But I know that the justification that allowed me to find the limit this way is not an analytic justification. How can I find this limit on an analytic basis?
Thanks
 [/INST] You know that $\lim \limits_{x \to +\infty} \left(\dfrac{x+\sin(x^2)}{\sqrt{x^2+1}}\right)= \lim \limits_{x\to +\infty}\left(\dfrac{x}{\sqrt{x^2+1}}+\dfrac{\sin(x^2)}{\sqrt{x^2+1}}\right)$.
Now if both $\lim \limits_{x\to +\infty}\left(\dfrac{x}{\sqrt{x^2+1}}\right)$ and $\lim \limits_{x\to +\infty}\left(\dfrac{\sin(x^2)}{\sqrt{x^2+1}}\right)$ exist, then $$ \lim \limits_{x\to +\infty}\left(\dfrac{x}{\sqrt{x^2+1}}+\frac{\sin(x^2)}{\sqrt{x^2+1}}\right)=\lim \limits_{x\to +\infty}\left(\frac{x}{\sqrt{x^2+1}}\right )+ \lim \limits_{x\to +\infty}\left(\frac{\sin(x^2)}{\sqrt{x^2+1}}\right)$$
You've proved that the first one on the RHS exists and equals $1$.
To show that the second one exists and equals $0$ it suffices to prove that $\lim \limits_{x\to +\infty}\left(\left | \dfrac{\sin(x^2)}{\sqrt{x^2+1}} \right |\right)=0$ and that's a consequence of the squeeze theorem because for all $x\in \mathbb{R},$ $$0\leq \left | \dfrac{\sin(x^2)}{\sqrt{x^2+1}} \right |\leq \left | \frac{1}{\sqrt{x^2+1}} \right |$$ and $\lim \limits_{x\to +\infty}\left(\left | \frac{1}{\sqrt{x^2+1}} \right |\right)=0$.
Therefore $$ \lim \limits_{x\to +\infty}\left(\dfrac{x}{\sqrt{x^2+1}}+\frac{\sin(x^2)}{\sqrt{x^2+1}}\right)=\lim\limits_{x\to +\infty}\left(\frac{x}{\sqrt{x^2+1}}\right )+ \lim\limits_{x\to +\infty}\left(\frac{\sin(x^2)}{\sqrt{x^2+1}}\right)=1+0.$$
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