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Evaluate $$\lim_{x \to \infty} \sqrt{x^2 + 2x} - x$$

Hello!

I was solving this problem, and here is my approach:

Inside the square root $x^2$ overpowers $2x$ as $x \to \infty$, so $$\lim_{x \to \infty} \sqrt{x^2 + 2x} - x = \lim_{x \to \infty} \sqrt{x^2} - x = \lim_{x \to \infty} |x| - x = 0$$ which is wrong answer. Actual answer is $1$.

This is the way I have solved so many problems when $x \to \infty$, especially when the terms are in denominator.

For example, consider $$\lim_{x \to \infty} \frac{x^2 + x}{2x^2 + 4x + 10}$$

Now, we ignore the smaller power terms, and answer is $\frac{1}{2}$. Why can we ignore smaller powers here but not in the original question on top?

Sorry if this is a stupid question.

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    $\begingroup$ Let me answer with a question: suppose we evaluate $\lim\limits_{x\to\infty}(x+1)-x$. Doesn't the $x$ overpower the $1$, so we get $\lim\limits_{x\to\infty}(x)-x=0$? $\endgroup$ Aug 5, 2022 at 17:19
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    $\begingroup$ Essentially, $x^2+2x$ is very close to $x^2+2x+1 = (x+1)^2$ (must closer to that than to $x$ for large $x$), so the square root is actually a lot closer to $x+1$ than to $x$. $\endgroup$ Aug 5, 2022 at 17:21
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    $\begingroup$ The way to think about your ratio example is to divide the numerator and denominator by $x^2$, so that$$\lim_{x\to\infty}\frac{x^2+x}{2x^2+4x+10}=\lim_{x\to\infty}\frac{1+\frac1x}{2+\frac4x+\frac{10}{x^2}}=\frac{1+0}{2+0+0}=\frac12.$$ $\endgroup$ Aug 5, 2022 at 17:21
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    $\begingroup$ No, you are not evaluating the limit. The limit is $+\infty$ or 'does not exist'. You are not evaluating the limit and then solving infinity - infinity, you are just using an approximation of $\sqrt{x^2 + x}$ that is not powerful enough. $\endgroup$
    – SBK
    Aug 5, 2022 at 17:28
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    $\begingroup$ @MangoPizza $$\sqrt{x^2 + 2x} = x(1 + 2/x)^{1/2} = x(1 + 1/x + \cdots) \approx x+1$$ $$(10^2 + 2 \times 10) \approx 10.95 \approx 10+1$$ $\endgroup$
    – DanielV
    Aug 5, 2022 at 17:32

4 Answers 4

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There is no mathematical notion of 'overpowering', so nobody can say precisely why you're idea about overpowering is wrong; it just isn't math unfortunately! It seems you're thinking in a way that is relying on too much 'intuition' about what is going to happen, before that intuition is actually developed enough to get the right answer. This is really common but you have to try to force yourself to use inequalities and mathematical arguments to get limits.

Also, once you know the answer and have proved it properly, you can try to come up with intuitive argument: $$ \sqrt{x^2 + 2x} = \sqrt{x(x+2)}, $$ which is the geometric mean of $x$ and $x+2$. Maybe for large $x$, since these numbers are relatively very close, this acts sufficiently like the arithmetic mean $$ \frac{x + x + 2}{2} = x+1. $$ So of course when you subtract $x$ you get 1.

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Let us try to eliminate or cancel the square term & the root term :

$\lim_{x \to \infty} {\sqrt{x^2 + 2x} - x} = \lim_{x \to \infty} {\sqrt{x^2 + 2x + 1 -1} - x} = \lim_{x \to \infty} {\sqrt{(x + 1)^2 -1} - x}$

We can now see that the square term will Over-Power the Constant.
We then get a way to eliminate or cancel the square term & the root term :

$\lim_{x \to \infty} {\sqrt{x^2 + 2x} - x} = \lim_{x \to \infty} {\sqrt{(x + 1)^2} - x} = \lim_{x \to \infty} {(x + 1) - x} = 1$

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$\lim_{x \to \infty} \sqrt{x^2+2x}-x =$
$\lim_{x \to \infty} (\sqrt{x^2+2x}-x) \cdot \frac{\sqrt{x^2+2x}+x}{\sqrt{x^2+2x}+x} =$
$\lim_{x \to \infty} \frac{(\sqrt{x^2+2x}-x)\cdot(\sqrt{x^2+2x}+x)}{\sqrt{x^2+2x}+x} =$
$\lim_{x \to \infty} \frac{|x^2+2x|-x^2}{\sqrt{x^2+2x}+x} =$
$\lim_{x \to \infty} \frac{x^2+2x-x^2}{\sqrt{x^2+2x}+x} =$
$\lim_{x \to \infty} \frac{2x}{\sqrt{x^2+2x}+x} =$
$\lim_{x \to \infty} \frac{2}{\frac{\sqrt{x^2+2x}}{x}+1} =$
$\lim_{x \to \infty} \frac{2}{\sqrt{1+\frac{2}{x}}+1} =$
$\frac{2}{\sqrt{1+\lim_{x \to \infty} \frac{2}{x}}+1} = 1$

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  • $\begingroup$ Avoid the use of $*$ to denote multiplication, that's a common practice in programming, not in Mathematics where it has other meanings. Use \cdot ($\cdot$), \times ($\times$) or (ideally) simply use juxtaposition. $\endgroup$
    – jjagmath
    Aug 5, 2022 at 17:45
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Okay.... I'd say the issue of "overpowering" is that it doesn't work when there isn't something else acting an the thing doing the overpowering to something finite, so that the thing being overpowered is forced to be insignificant. If the thing doing the overpowering goes to infinity and is only countered by something else going to negative infinity so they cancel out, then the thing being overpowered isn't being forced to go to $0$-- it's merely going ... somewhere else. Just not as fast as the overpowering thing.

As $x\to \infty$ then the $\sqrt{x^2} \to \infty$ and a higher rate than $\sqrt{+2x} \to \infty$ but that doesn't mean it goes to $0$.

We have $\sqrt{x^2 + 2x}> \sqrt {x^2} = x$ so we must have $\sqrt{x^2+2x}-x = f(x) > 0$.

Now we do know that $f(x) << x$

and as $x\to SomethingREALLYhuge$
we will have $f(x)\to SomethingNOWHEREnearashuge$,

but there is nothing forcing it to zero. We have no idea at this point what $\lim_{x\to\infty}\sqrt{x^2 + 2x} -x = \lim_{x\to \infty} f(x)$ ought to be.

[Compare this to $\frac {x^5 + x^3}{2x^5 + 4x^2}$ where the $x^5$s "overpower" then $x^3$ and $4x^2$ and breezily say $\lim \frac {x^5 + x^3}{2x^5 + 4x^2}\approx \lim \frac {x^5}{2x^5} = \frac 12$. In this case we are dividing the $x^5$ by $x^5$ to "overpower to something finite". So as $x^5$ is being forced to something finite the $x^3$ and $x^2$ are forced to $\frac {finite}{x^2}$ and $\frac {finite}{x^3}$ and go to zero.]

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Now to actually do this thing. $\sqrt{x^2+2x} = \sqrt{x^2 + 2x + 1 -1}= \sqrt{(x+1)^1 - 1}$.

Now we can say that as $x\to \infty$ that $(x+1)^2$ "overpowers" the $-1$. The difference between $x^2$ overpowering $-1$ to insignificance and $x^2$ overpowering $2x$ but not to insignificance, is that $-1$ is finite and gets insignificant when compared to very large $x$, whereas $2x$ is not finite.

$\lim \sqrt{x^2 + 2x} -x =\lim \sqrt{(x+1)^2-1} - x\approx \lim \sqrt{(x+1)^2} - x=\lim (x+1) -x = 1$.

.....

Still, informality is only for estimating and intuiting answers. To actually answer the question we must be more formal and proper.

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