Question

find limit as x approaches 0 of (e^x + 5x)^(2/x)

Answer 1

so i got the indeterminate form of 1^infinity what should ido next?

Answer 2

\[\Large\rm \lim_{x\to0}(e^x+5x)^{2/x}\]Okkayyyyy.
I guess we should logs.\[\LARGE\rm e^{\left[\ln\lim_{x\to0}(e^x+5x)^{2/x}\right]}\]

Answer 3

Does that step make sense or not so much?
If it's easier, we can write a "left side" as y,
and then take the log of both sides.

Answer 4

This is a good trick to get comfortable with though

Answer 5

yes please continue :D

Answer 6

Ignoring the base e for a moment,
let's see what's happening up in the exponent.\[\Large\rm \ln\lim_{x\to0}(e^x+5x)^{2/x}\]Pass the log into the limit,\[\Large\rm \lim_{x\to0}\ln\left[(e^x+5x)^{2/x}\right]\]Applying rules of logs allows us to bring the 2/x outside,\[\Large\rm \lim_{x\to0}\frac{2}{x}\ln(e^x+5x)\]

Answer 7

Let's write the x in the denominator, it might give us one of the indeterminate forms that we can work with,\[\Large\rm 2\lim_{x\to0}\frac{\ln(e^x+5x)}{x}\]

Answer 8

So what is our stuff approaching now? :)

Answer 9

u mean to the left or to right?

Answer 10

Err I guess what I mean is....
Just plug in x=0 and see what indeterminate form we're getting.
We should still be getting an indeterminate form.

Answer 11

but x in the bottom is 0

Answer 12

So the bottom is approaching 0.
What is the top approaching?

Answer 13

0 too?

Answer 14

ln(1) = 0

Answer 15

Ok good!
So we're getting the indeterminate form 0/0.

Which is one that allows us to use our old friend \(\Large\rm L'Hospital!\)

Answer 16

derivative right

Answer 17

Yes derivative of top and bottom, separately.

Answer 18

for top I got (e^x +5) / (5x + e^x)
for bottom I got 1/x^2

Answer 19

is this right

Answer 20

Woops let's fix the bottom a sec.
The bottom is x.
`It is not` 1/x.

Answer 21

yeah i keep make mistake over and over

Answer 22

>.< heh

Answer 23

so its 1

Answer 24

Ok great, so we'll simplify the fraction down,
\[\Large\rm 2\lim_{x\to0}\frac{e^x+5}{e^x+5x}\]

Answer 25

So what are we getting now?
Still indeterminate form or no?

Answer 26

12

Answer 27

Good!
And remember that we had that base of e right??

Answer 28

oh yeah

Answer 29

e^12

Answer 30

Yay good job \c:/

Answer 31

wohoho thank you so much

Answer 32

np :3