Graphing f(x) = e^x/x and Testing for Infinity

[tomwilliam]

When sketching a graph I'm told to assume that the expression:

f(x) =( e^x)/x

Tends towards the infinite as x tends towards the infinite. Can someone show me how to check this?

Thanks

 

[Square]

The short answer is that the exponential function $a^x$ increases faster than any power of $x$ ($x^{\alpha}, \ \alpha \in \mathbb{R}$).

The long answer is that you could prove that the limit $\displaystyle \lim_{x\to\infty} \frac{a^x}{x^{\alpha}}$ (and thus that your given limit tends towards inf) tends towards infinity for any $a > 0$ and $\alpha \in \mathbb{R}$ by expanding $a^x = (1+p)^x \geq (1+p)^n$, where $p > 1$ and $n$ is the integer part of $x$, and then doing some algebra. You should have come up with an expression which is smaller than $\frac{a^x}{x^{\alpha}}$ which tends to infinity, which implies the wanted result.

 

[tomwilliam]

Thanks, very clear.