Question

How do I find the limit as x approaches negative infinity of x^3e^x?

I know that I am supposed to use L'Hopital's rule, but I'm not sure how

Answer 1

\[\lim_{x \rightarrow - \infty} \space ^{x ^{3 \space e^x}} \]

Answer 2

Do you have any idea on how to approach this problem?

Answer 3

Youu are correct about using L Hospital rule
However, to use L Hospital rule, we will convert our function into the fraction form

Answer 4

\[= \lim_{x \rightarrow \infty} \space \frac{ x ^{3} }{ e ^{-x} }\]

Answer 5

if I bring e^x to the bottom, its exponent will change sign right?

Answer 6

Now let's take derivative of both top and bottom, would you like to give it a try?

Answer 7

Yeah I tried that, and I got 3x^2/-e^-x

Answer 8

Excellent!

Answer 9

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Answer 10

\[\lim_{x \rightarrow \infty} \space \frac{ 3x ^{2} }{ -e ^{-x}}\]

Answer 11

If we evaluate this limit when x approaches negative infinity, do you have any idea on what this fraction approaches?

Answer 12

Infinity over -infinity?

Answer 13

That's right :)

Answer 14

So it is still indeterminate form right?

Answer 15

Yeah so I did the rule twice more and got 6/-e^-x, but that didn't really work

Answer 16

Great! It looks good to me

Answer 17

\[= \lim_{x \rightarrow \infty} \space \frac{ 6 }{ -e ^{-x} }\]

Answer 18

Now we don't have indefinite form anymore because the top is fixed at 6
what will the bottom approach?

Answer 19

-infinity?

Answer 20

Exactly

Answer 21

if the bottom is huge and the top is small, then what does the whole fraction approach?

Answer 22

OHH I see it now! It's zero. I forgot about this property, haha. Thank you so much!

Answer 23

Haha, no worries. =]

Answer 24

uhh

Answer 25

why did u say the bottom is going to infinity?

Answer 26

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Answer 27

there is some error in the solution method

Answer 28

i think what u want to write is this