It will hardly take 1 second

Calculus Level 1

Detemine :
$\large \displaystyle\lim_{n\rightarrow \infty} \dfrac{ e^n }{n^2}$ .

Details : e is an exponential function .

\infty

e 2 0

2 solutions

Brock Brown
Sep 28, 2015

By L'Hopital's Rule , we can say...

$L = \lim_{n \to \infty} \frac{e^n}{\color{#D61F06}{n^2}} = \frac{\infty}{\infty} \implies$

$L = \lim_{n \to \infty} \frac{e^n}{\color{#D61F06}{2n}} = \frac{\infty}{\infty} \implies$

$L = \lim_{n \to \infty} \frac{e^n}{\color{#D61F06}{2}} = \frac{\infty}{2} \implies$

$L = \boxed{\infty}$

Useful tip. Try letting m = 1/n, so you don't have to deal with infinities.

Hint: Show that e^x is approximately equals to x for small x (when it's close to 0).

By the way, it is a common convention to hate on Lhopital rule because it is too cheesy of a solution. It's like going into a classroom and deciding to hate the 1st person you meet. In this case, L'hopital is that student.

Pi Han Goh - 5 years, 8 months ago

Otto Bretscher
Sep 12, 2015

$e^n>\frac{n^3}{3!}$ for positive $n$ , by Taylor Series, so $\frac{e^n}{n^2}>\frac{n}{6}$ and $\lim_{n\to\infty}\frac{e^n}{n^2}=\boxed{\infty}$

Sir can we say something like this :

$\frac { e^{ \infty } }{ \infty \infty } =\frac { 1 }{ \infty } =0$

Syed Baqir - 5 years, 9 months ago

Wrong. it is infinity/infinty = undefined.

Pi Han Goh - 5 years, 9 months ago

Hmm, Thanks for reply :)

Syed Baqir - 5 years, 9 months ago

I also thought

Turbulent Payel - 5 years, 7 months ago

It will hardly take 1 second

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