∞ * 0

>>8846817
Tip:
[math]e^\alpha = \frac{1}{e^{-\alpha}}[/math]

You should have learned this in highschool but I guess you actually dont know what to do.

Search for "rule of l'hospital" and you should be able to figure out the rest.

Exponentials beat polynomials

>>8846831
Im in highschool, my teachers hasnt touched this subject. She's pretty old, doesnt have the teaching capabilitiies she once had

It suffices to show that after one point on the negative axis, [math] \alpha \cdot e^{\alpha} [/math] is squeezed in between the zero line and [math] \frac{1}{\alpha} [/math]

i.e

[math] \alpha \cdot e^{\alpha} > \frac{1}{\alpha} [/math]

i.e.

[math] \alpha^2 \cdot e^{\alpha} < 1 [/math]

In reality this even holds on the whole negative axis.

>>8846833
Did you look up l'Hospital?

Also if youre underage b& please do get off this site. Wether you see it or not, you are bringing down what little quality this board has left.

>>8846840
And that's true because the function has a global maximum on that interval, at [math]\alpha=-2[/math],
where it evaluates to
[math] 4 e^{-2} < 1 [/math]

>>8846831
to be fair, sometimes highschools / unis want their students to solve certain limits without the l'hospital rule in order for them to grasp the principles of limits better.
then it introduces that rule after a few weeks or so, to then move on to more difficult problems

File: 1492827298874.gif (569KB, 362x366px) Image search: [Google]

569KB, 362x366px

>>8846817
USE L'HÔPITAL

[eqn]
\lim\limits_{\alpha \rightarrow -\infty} \alpha e^\alpha = \lim\limits_{\alpha \rightarrow -\infty} \frac{\partial}{\partial \alpha}\frac{\alpha}{e^{-\alpha}} = \lim\limits_{\alpha \rightarrow -\infty}\frac{1}{e^{-\alpha}} = 0
[/eqn]

>>8846870
then your teacher most likely wants you to solve the problem without it, it's more of an intuitive question.

>>8846817
https://math.stackexchange.com/questions/767781/why-does-lim-limits-t-to-inftytet-0
there's a proof at the bottom

>>8846817
Looks like an L not an e.

Lim alpha * e^alpha =
Lim alpha * Lim e^alpha
= 0 * infinity = 0

>>8846921
dude infinity isn't element of the real numbers

>>8846921
>0 * infinity = 0
No.

>>8846924
>>8847328
I know you can't actually break them up like that, but I doubt op knows how to do delta epsilon proofs.