How to find an analytic solution to $\lim_{x \to +\infty} \frac{x+\sin(x^2)}{\sqrt{x^2+1}}$

Question

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

Answer 1

Write, for $x>0$ $${x+\sin(x^2)\over\sqrt{x^2+1}}= {{1\over x}\cdot(x+\sin(x^2))\over{1\over x}\sqrt{x^2+1}}= {{1+{\sin(x^2)\over x} } \over\sqrt{1+{1\over x^2}}}.$$

Answer 2

You know that $\displaystyle \lim_{x \to +\infty} \space \frac{x+sin(x^2)}{\sqrt{x^2+1}}= \lim_{x\to +\infty}\Bigl(\frac{x}{\sqrt{x^2+1}}+\frac{sin(x^2)}{\sqrt{x^2+1}}\Bigr)$, now if both $\displaystyle \lim_{x\to +\infty}\Bigl(\frac{x}{\sqrt{x^2+1}}\Bigr )$ and $\displaystyle \lim_{x\to +\infty}\Bigl(\frac{sin(x^2)}{\sqrt{x^2+1}}\Bigr)$ exist, then $$ \lim_{x\to +\infty}\Bigl(\frac{x}{\sqrt{x^2+1}}+\frac{sin(x^2)}{\sqrt{x^2+1}}\Bigr)=\lim_{x\to +\infty}\Bigl(\frac{x}{\sqrt{x^2+1}}\Bigr )+ \lim_{x\to +\infty}\Bigl(\frac{sin(x^2)}{\sqrt{x^2+1}}\Bigr)$$

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 $\displaystyle \lim_{x\to +\infty}\Bigl(\biggl | \frac{sin(x^2)}{\sqrt{x^2+1}} \biggr |\Bigr)=0$ and that's a consequence of the squeeze theorem because for all $x\in \mathbb{R}$ $$0\leq \biggl | \frac{sin(x^2)}{\sqrt{x^2+1}} \biggr |\leq \biggl | \frac{1}{\sqrt{x^2+1}} \biggr |$$ and $\displaystyle \lim_{x\to +\infty}\Bigl(\biggl | \frac{1}{\sqrt{x^2+1}} \biggr |\Bigr)=0$.

Therefore $$ \lim_{x\to +\infty}\Bigl(\frac{x}{\sqrt{x^2+1}}+\frac{sin(x^2)}{\sqrt{x^2+1}}\Bigr)=\lim_{x\to +\infty}\Bigl(\frac{x}{\sqrt{x^2+1}}\Bigr )+ \lim_{x\to +\infty}\Bigl(\frac{sin(x^2)}{\sqrt{x^2+1}}\Bigr)=1+0$$

Answer 3

If $h(x)\leq f(x)\leq g(x) \forall x\in \Bbb R\implies \lim_{x\to\infty}h(x)\leq \lim_{x\to\infty}f(x)\leq \lim_{x\to\infty}g(x)$

Here, $$h(x)=\frac{x-1}{\sqrt{x^2+1}},f(x)=\frac{x-\sin(x^2)}{\sqrt{x^2+1}})$$ and $$g(x)=\frac{x+1}{\sqrt{x^2+1}}$$