Question

Use L'hospital rule where applicable

lim √x(e^x/2)
x->∞

Answer 1

Everything under the root, or just x?

Answer 2

just x

Answer 3

root x time e^(-x/2)

Answer 4

As \(\Large x\to\infty\), our function is approaching the indeterminate form \(\Large \infty\cdot0\) right?

Answer 5

right

Answer 6

Using rules of exponents,
\[\Large \lim_{x\to\infty}\sqrt x \;e^{-x/2}\quad=\quad \lim_{x\to\infty}\frac{\sqrt x}{e^{x/2}}\]

Answer 7

Now it's giving us the indeterminate form \(\Large \dfrac{0}{0}\).
Good! Now it's in a nice form where we can apply L'Hop, yes?

Answer 8

Sorry I mean infty/infty*

Answer 9

yeah thats as far as i got on my own. im not 100% confident where to go from there

Answer 10

So applying L'Hop,\[\Large L'H\quad=\quad \lim_{x\to\infty}\frac{(\sqrt{x})'}{\left(e^{x/2}\right)'}\]What derivatives do we get? :o
Remember the derivative of sqrt x?

Answer 11

yeah its x^1/2

Answer 12

So that's sqrt x rationalized :D But what is its derivative?

Answer 13

Power rule, yes?

Answer 14

1/2 x ^-1/2

Answer 15

Ok good! We'll write it like this,\[\Large \left(\sqrt x\right)'\quad=\quad \frac{1}{2\sqrt x}\]so it's a bit easier to work with.

Answer 16

So our derivatives should look like this:\[\Large L'H\quad=\quad \lim_{x\to\infty}\frac{(\sqrt{x})'}{\left(e^{x/2}\right)'}\quad=\quad \lim_{x\to\infty}\frac{\left(\dfrac{1}{2\sqrt x}\right)}{\left(\frac{1}{2}e^{x/2}\right)}\]

Answer 17

next step would be to multiply by 1/2 correct?

Answer 18

Multiply the top and bottom by 2? Sure! That might clean things up a tad.

Answer 19

We can also bring the sqrtx down into the denominator.\[\Large \lim_{x\to\infty}\frac{-1}{\sqrt x\;e^{-x/2}}\]Woops I missed a negative when I took the derivative of the exp part. Fixed it.

Answer 20

Hmmm that doesn't seem to have helped us much -_- hmmmmm

Answer 21

what i did was \[1/2\sqrt{x *e^x}\]

Answer 22

Oh I didn't miss a negative did I?
It became positive when we brought it into the denom* blah.

Answer 23

and then plug in the infinity to get 1/2* infinite * infinite = 1/infinite = 0

Answer 24

is that correct

Answer 25

\[\Large \lim_{x\to\infty}\frac{1}{\sqrt {x\;e^{x}}}\]

Answer 26

I think you have an extra factor of 1/2 in there, but ya looks like you've got it figured out.
For some reason I was thinking infty * infty was an indeterminate form :)

Answer 27

0 is correct, yay good job!

Answer 28

thanks for guiding me