Question

How do I calculate the limit of x*exp(-x) ?

Answer 1

as x approaches what value?

Answer 2

as x approaches + infinity?

Answer 3

Going to infinity.

Answer 4

I was trying to take the derivative of this product but I can't get it right.

Answer 5

You don't "calculate" it. You find it or determine it.

Answer 6

@tkhunny Okay, that's right. I didn't express myself properly. So, I was trying to apply the l'Hopital rule but it seems to complicate the matters.

Answer 7

e^-x controls the function for large x doesnt it?

Answer 8

x/e^x

e^x grows much faster than x lending us to conclude with a high degree of certainity that this goes to 0

Answer 9

@amistre64 Well, yes. The exponent vanishes faster. I thought there is some more general explanation. I mean, what if I had x^4*exp(-x)?

Answer 10

e^x still grows faster

Answer 11

my initial idea was the taylor polynomial

Answer 12

x * e^-x = e / e^x differentiate top and bottom of fraction gives you 1 / e^x

Answer 13

l'hopital rule should do it - if i've interpreted the problem correctly

Answer 14

typo x/e^x not e / e^x

Answer 15

lhop is for 0/0, can it be applied to inf/inf?

i recall my prof insisting the 0/0 criteria

Answer 16

@amistre64 We've applied it to that case, as well.

Answer 17

yes it can be applied to inf/inf

Answer 18

so the limit is 0

Answer 19

@cwrw238 Yes, thanks. Got it.

Answer 20

yw

Answer 21

How about the case \[x^4\exp(-x)\]. If I find the derivative, I get this: \[4x^3 exp(-x) - x^4 exp(-x) \]. So, it turns out to be recursive as far as I see.

Answer 22

yes - not sure about that one

Answer 23

if you keep on differentiating it would be pretty complex!!

Answer 24

Yes, I noticed. Indeed, that's why I posted the question. I can't just determine intuitively which increases/decreases faster.

Answer 25

\[\large \dfrac{x^{1000000000}}{e^x}\]

Answer 26

Notice that after applying L'Hopital's rule a billion times, the numerator vanishes but the denominator doesn't move an inch. e^x is e^x forever

Answer 27

ah!!

Answer 28

so the limit is 0

Answer 29

So, how do I know at which point the numerator wins over the exponent?

Answer 30

\[\large \large \dfrac{x^{1000000000}}{e^x} \leadsto \dfrac{c}{e^x}\]

Answer 31

so simple really ...

Answer 32

OS is dancing on my side, i see all replies jumbled up..