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I'm a Calculus I student and my teacher has given me a set of problems to solve with L'Hoptial's rule. Most of them have been pretty easy, but this one has me stumped.

$$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}$$

You'll notice that using L'Hopital's rule flips the value of the top to the bottom. For example, using it once returns:

$$\lim\limits_{x\to \infty} \frac{\sqrt{x^2 + 1}}{x}$$

And doing it again returns you to the beginning:

$$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}$$

I of course plugged it into my calculator to find the limit to evaluate to 1, but I was wondering if there was a better way to do this algebraically?

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    $\begingroup$ $\frac{x}{\sqrt{x^2 + 1}}=\frac{\sqrt{x^2}}{\sqrt{x^2+1}}=\sqrt{\frac{x^2}{x^2+1}}$ $\endgroup$ – Mason Jan 5 at 18:41
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    $\begingroup$ The "right" way to solve this is as Thomas Shelby wrote in his answer. However, you can almost intuitively see that the limit is $1$ even without using a calculator. Indeed, if $x$ is a huge positive number, say $10^{5863}$, the difference between $x^2$ and $x^2 + 1$ is extremely tiny. So the expression is essentially indistinguishable from ${x\over\sqrt{x^2}} = {x\over x} = 1$. $\endgroup$ – Andreas Rejbrand Jan 5 at 19:37
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Hint: Divide the numerator and denominator by $x $ and apply the limit.

$$\frac{x}{\sqrt{x^2 + 1}}=\frac{1}{\sqrt{1 + \frac{1}{x^2}}}$$

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    $\begingroup$ In other words, actually do the division you see written down in $\lim_{x\to\infty}\frac{\sqrt{x^2+1}}x$. $\endgroup$ – Arthur Jan 5 at 18:40
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Hint

Simply use $${x\over x+1}={x\over \sqrt{x^2+2x+1}}<{x\over \sqrt{x^2+1}}<1$$for large enough $x>0$.

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    $\begingroup$ Mostafa.Very nice+. $\endgroup$ – Peter Szilas Jan 5 at 19:01
  • $\begingroup$ But how so we know the limit is 1. Based on what you wrote couldnt the answer be 1/2 $\endgroup$ – Milan Jan 5 at 19:05
  • $\begingroup$ @PeterSzilas thank you! $\endgroup$ – Mostafa Ayaz Jan 5 at 19:08
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    $\begingroup$ @MilanStojanovic No since $$\lim_{x\to \infty}{x\over x+1}=1$$and $$\lim_{x\to \infty}1=1$$so the Squeeze theorem is applicable here.... $\endgroup$ – Mostafa Ayaz Jan 5 at 19:09
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By your own reasoning, you have the following: $$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}=\lim\limits_{x\to \infty} \frac{\sqrt{x^2 + 1}}{x}$$

Now, the left side is clearly the reciprocal of the right side, so we have: $$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}=\frac{1}{\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}}$$

(Note that doing this manipulation assumes that $\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}$ converges to a real number. However, you can use the first derivative to show this is an always increasing function and then use basic algebra to show that $\frac{x}{\sqrt{x^2 + 1}} < 1$ for all $x\in\Bbb{R}$. Thus, because this is a bounded, always increasing function, the limit as $x\to \infty$ must converge to some real number, so our assumption in this manipulation is valid.)

Cross-multiply: $$\left(\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}\right)^2=1$$

Take the square root: $$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}=\pm 1$$

However, it is easy to show that $\frac{x}{\sqrt{x^2 + 1}}>0$ for all $x > 0$. Therefore, there's no way the limit can be a negative number like $-1$. Thus, the only possibility we have left is $+1$, so: $$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}=1$$

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When computing the limit of rational functions, as is the case for $$\lim_{x \rightarrow \infty} \frac{x}{\sqrt{x^2 +1}},$$ you want to divide the top and bottom by the highest degree in the denominator, which in this case is $x$. Since $x \rightarrow +\infty$, so $x$ is always positive (at least, near where we are worried about) I claim that $x = \sqrt{x^2}$. So, if we divide the top and bottom by $x$, we get $$\lim_{x \rightarrow \infty} \frac{x}{\sqrt{x^2 +1}} = \lim_{x \rightarrow \infty} \frac{1}{\sqrt{1 + 1/x^2}}.$$ You should be able to compute the limit from here.

Whenever you see a monomial in the numerator with the square root of a polynomial in the denominator, you should consider this method. Of course, keep in mind that you'll have to tweak it slightly if $x \rightarrow -\infty$! Try to see if you can figure out what would change in that case.

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Set $x = \sinh t$. We have $$\frac{x}{\sqrt{x^2+1}}= \frac{\sinh t}{\sqrt{1+\sinh^2t}} = \frac{\sinh t}{\cosh t} = \tanh t$$

$x \to \infty$ is equivalent to $t\to\infty$ so $$\lim_{x\to\infty} \frac{x}{\sqrt{x^2+1}} = \lim_{t\to\infty} \tanh t = \lim_{t\to\infty}\frac{e^t - e^{-t}}{e^t+e^{-t}} = \lim_{t\to\infty}\frac{e^{2t}-1}{e^{2t}+1} = 1$$

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