Intuition for $\lim_{x\to\infty}\sqrt{x^6 - 9x^3}-x^3$

Question

Trying to get some intuition behind why: $$ \lim_{x\to\infty}\sqrt{x^6-9x^3}-x^3=-\frac{9}{2}. $$

First off, how would one calculate this? I tried maybe factoring out an $x^3$ from the inside of the square root, but the remainder is not factorable to make anything simpler. Also tried expressing $x^3$ as $\sqrt{x^6}$, but that doesn't really help either.

One would think that, as $x^6$ grows more quickly than $x^3$ by a factor of $x^3$, the contribution of the $x^3$ term to the term in the square root would be dwarfed by the contribution of the the $x^6$ term, so the overall behavior of the first term in the limit would "behave" like $x^3$, as x gets bigger and bigger, so I would think intuitively that the limit would evaluate to 0.

Answer 1

Well,

$$\sqrt{x^6 - 9 x^3} = x^{3} \left(1 - \frac{9}{x^{3}}\right)^{1/2}$$

and so the question is really about understanding how $\sqrt{1 - t}$ looks when $t$ is pretty small. By noticing that the derivative $\frac{d}{dt} \sqrt{t} = \frac 1 2$ when $t = 1$, we can say that

$$\left(1 - \frac{9}{x^{3/2}}\right)^{1/2} \approx \frac 1 2 \left(\frac{-9} {x^{3}}\right)$$

which leads quickly to the claimed limit of $-9/2$.

---

Alternatively, there is a completely standard technique of multiplying and dividing by the conjugate, which is $\sqrt{x^6 - 9x^3} + x^3$, but I find that this isn't very enlightening.

Answer 2

$$\sqrt{x^6-9x^3}-x^3=\frac{x^6-9x^3-x^6}{\sqrt{x^6-9x^3}+x^3}=\frac{-9x^3}{\sqrt{x^6-9x^3}+x^3}$$

As $x$ goes to $\infty$, $\sqrt{x^6-9x^3} \approx x^3.$

More, precisely, divide the numerator and denominator by $x^3$:

\begin{align} \sqrt{x^6-9x^3}-x^3&=\frac{x^6-9x^3-x^6}{\sqrt{x^6-9x^3}+x^3}\\&=\frac{-9x^3}{\sqrt{x^6-9x^3}+x^3} \\ & =\frac{-9}{\sqrt{1-9x^{-3}}+1} \end{align}

Answer 3

Hmmm, no one has pointed out the obvious:

$\lim \sqrt{x^6 - 9x^3} - x^3 = \lim \sqrt{x^6 - 9x^3 + 36/4} - x^3$

$ = \lim \sqrt{(x^3 - 9/2)^2} - x^3 = \lim x^3 - 9/2 - x^3 = -9/2$

Intuition.... hmm .... I guess realizing the $x^3$ from $\sqrt {x^6 + stuff}$ was going to cancel the $-x^3$. So I want some $\lim \sqrt {Y_{x^3}^2} - x^3$ and figuring $Y_{x^3}^2$ must be.

=====

So why does $\lim \sqrt{x^6 -9x^3} = \lim\sqrt{x^6 -9x^3 + 36/4}$?

Let $\epsilon > 0$. We wish to solve for which $x$ we have $|\sqrt{x^6 -9x^3 + 36/4}-\sqrt{x^6-9x^3}| = x^3 - \frac 92 - \sqrt{x^6-9x^3} < \epsilon$. If we can show that this can be solved for all $x > M$ for some $M$ we are done.

$(x^3 - \frac 92) - \epsilon < \sqrt{x^6 - 9x^3} $. For large enough $x$ we may assume this are both positive.

$x^6 - 9x^3 + 9 - 2\epsilon*(x^3 - \frac 92) + \epsilon^2 < x^6 - 9x^3$

$9 - 2\epsilon*(x^3 - \frac 92) + \epsilon^2 < 0$

$(x^3- \frac 92) > \frac{(9 + \epsilon^2)}{2\epsilon}$

So for any $x > \sqrt[3]{\frac 92 + \frac{(9 + \epsilon^2)}{2\epsilon}}$ this will be true.

So for all $\epsilon > 0$ if $x > M = \sqrt[3]{\frac 92 + \frac{(9 + \epsilon^2)}{2\epsilon}}$ we have $|\sqrt{x^6 -9x^3 + 36/4}-\sqrt{x^6-9x^3}| < \epsilon$

So $\lim_{x\rightarrow \infty}\sqrt{x^6 -9x^3 + 36/4}-\sqrt{x^6-9x^3}= 0$

So $\lim_{x\rightarrow \infty}\sqrt{x^6 -9x^3 + 36/4}= \lim_{x\rightarrow \infty}\sqrt{x^6 -9x^3}$

So $\lim_{x\rightarrow \infty}\sqrt{x^6 -9x^3}-x^3 = \lim_{x\rightarrow \infty}\sqrt{x^6 -9x^3 + 36/4} -x^3 = -\frac 92$

Or more generally...

If $c_x \rightarrow \infty$ and $f$ is continuous, then $\lim_{x\rightarrow \infty}f(c_x + h) = \lim_{x\rightarrow \infty}f(c_x(1 + h/c_x))=\lim_{x\rightarrow \infty}f(c_x(\lim_{x_\rightarrow \infty}(1 + h/c_x))=\lim_{x\rightarrow \infty}f(c_x*1)=\lim_{x\rightarrow \infty}f(c_x)$

Let $c_x = x^6 - 9x$, $h= 9=36/4$ $f(y) = \sqrt{y}$.

Answer 4

Its best to have a look at this answer to a similar question. Apart from that I want to add that most of the stuff in calculus/real-analysis is not to be handled by intuition. Even mathematicians have been led astray with their intuition in real-analysis (there exist continuous everywhere but differentiable nowhere functions, there exist differentiable functions whose derivatives are not integrable in sense of Riemann).

In particular the concept of limit is also not as intuitive as one might think of (i.e. on a level of $+, -, \times, /$). A limit is always evaluated using theorems which deal with evaluation of limits and nothing else. And worst approach to limits is to think of them as an approximation and hence $\sqrt{x^{6} - 9x^{3}}$ can't be replaced by $x^{3}$ because they are approximately equal. The simplest and straight-forward approach to such simple limit questions is the use of standard limits. Here we use the standard limit $$\lim_{x \to a}\frac{x^{n} - a^{n}}{x - a} = na^{n - 1}\tag{1}$$ First we replace $x^{3}$ by $1/h$ and as $x \to \infty$ we have $h \to 0^{+}$. Then \begin{align} L &= \lim_{x \to \infty}\sqrt{x^{6} - 9x^{3}} - x^{3}\notag\\ &= \lim_{h \to 0^{+}}\sqrt{\frac{1}{h^{2}} - \frac{9}{h}} - \frac{1}{h}\notag\\ &= \lim_{h \to 0^{+}}\frac{\sqrt{1 - 9h} - 1}{h}\notag\\ &= \lim_{t \to 1^{-}}\dfrac{t^{1/2} - 1}{\dfrac{1 - t}{9}}\text{ (putting }1 - 9h = t)\notag\\ &= -9\lim_{t \to 1^{-}}\frac{t^{1/2} - 1}{t - 1}\notag\\ &= -9 \cdot\frac{1}{2}\text{ (using (1))}\notag\\ &= -\frac{9}{2}\notag \end{align}

Answer 5

The only thing I would add to T. Bongers' answer is that the key to these kinds of questions is generally to ask:

1. How can I transform the limit to evaluate it using standard techniques?
2. Especially with limits at infinity, what is 'dying off' quicker?

Answer 6

For a fixed positive $k$ we should expect $\sqrt {y+k}$ to be very close to $\sqrt y$ when $y$ is very large, because when $d$ is not close to $0,$ we expect a big difference between $(d+\sqrt y)^2$ and $y,$ which means $d+\sqrt y$ is too big to be $\sqrt {y+k}$.

We can confirm this by observing that if $k,y$ are positive and $\sqrt {y+k}=\sqrt y+e_y$ then $$y+k=(\sqrt y +e_y)^2=y +2e_y\sqrt y +e_y^2>y+2e_y\sqrt y$$ which implies $0<e_y<k/2\sqrt y.$

With $y=x^6-9x^3$ and $k=(9/2)^3,$ we have $x^3-9/2=\sqrt {y+k}=e_y+\sqrt y.$

Answer 7

For intuition, first get rid of the annoying cubes,

$$\lim_{x\to\infty}\sqrt{x^6 - 9x^3}-x^3=\lim_{x\to\infty}\sqrt{x^2 - 9x}-x.$$

Then the classical trick is to multiply/divide by the conjugate binomial and simplify,

$$\lim_{x\to\infty}\left(\sqrt{x^2 - 9x}-x\right)\frac{\sqrt{x^2 - 9x}+x}{\sqrt{x^2 - 9x}+x}=\lim_{x\to\infty}\frac{-9x}{\sqrt{x^2 - 9x}+x}=-\frac92.$$

Another approach is to factor out $x$ and use the Taylor development $\sqrt{1+t}\approx 1+t/2$:

$$\lim_{x\to\infty}x\left(\sqrt{1 - \frac9x}-1\right)=\lim_{x\to\infty}x\left(- \frac9{2x}\right).$$

---

You can "feel" why the answer is not $0$ by comparing the plots of the functions $\sqrt{x^2-9x}$ and $x$.

Answer 8

T. Bongers' answer is right, but a bit sloppy. A better way would be to use the binomial expansion.

$$\sqrt{1-9x^{-3}} = \sum_{n=0}^\infty \frac{(1/2)^{\color{red}{n}}(-9)^nx^{-3n}}{n!} = 1 - \frac{9}{2x^{3}} - \frac{243}{8x^{6}} - \cdots$$ Where the red thing represents a falling power. $$\lim_{x\to+\infty}\sqrt{x^6-9x^3}-x^3=\lim_{x\to+\infty}x^3(\sqrt{1-9x^{-3}} - 1) = \lim_{x\to+\infty}x^3\left(- \frac{9}{2x^{3}} - \frac{243}{8x^{6}} - \cdots\right) = -\frac{9}{2}$$