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Here is a limit that bugged me for more than three or four days:

$$\lim_{x\rightarrow0}\dfrac x{x^3}=\infty.$$

How is that possible? I would think that this goes to zero but the answer is infinity, how?

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What happens when you simplify the fraction? – Ali Caglayan Feb 19 '14 at 20:35
$x/x^3 = 1/x^2$. – AnonSubmitter85 Feb 19 '14 at 20:35
up vote 6 down vote accepted

As $x\to0$, $x$ goes to zero, but $x^3$ will tend to zero faster. You may have the false notion that $x^3\gt x$ because of comparing big numbers. But as we approach $0$, $x^3$ will start to become smaller and smaller than $x$, starting from $1$ as the following graph shows:

$\phantom{kkkkXX}$Plot made by your humble correspondent Hakim ;-)

Furthermore, we can cancel out one $x$ to have a $1/x^2$ which obviously and intuitively tends to positive infinity. $($ since $x^2\geqslant0)$
We can illustrate this case using L'Hopital's rule since we have a $0/0$ case: $$\lim\limits_{x\to0}\frac{x}{x^3}=\lim\limits_{x\to0}\frac1{3x^2}=\ldots$$ Finally, the graph of $x/x^3$:

$\phantom{XkkkkXX}$Plot made by your humble correspondent Hakim ;-)

I hope this helps.
Best wishes, $\mathcal H$akim.

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thank you man! appreciate your effort! – limitsintermsofx Feb 19 '14 at 20:51
@limitsintermsofx You're welcome, glad it helps. – Hakim Feb 19 '14 at 20:51

What is $$\frac1{(.1)^2}?$$ $$\frac1{(.01)^2}?$$ $$\frac1{(.001)^2}?$$ $$...$$ $$\frac1{(.000000000000001)^2}?$$ HINT: $$\frac1{(.1)^2}=\frac1{\frac1{10^2}}=100$$

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HINT : $$\lim_{x\to0}\frac{x}{x^3} = \lim_{x\to 0}\frac{1}{x^2}$$

Can you handle from here?

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