Question

Use L'Hopitals rule
limx→0+ (integral from 0 to x of sqrt(t)cos(t)dt)/x^2
The answer is infinity, could someone explain it to me?

Answer 1

well applying L'hospital rule, what do you get?

Answer 2

The integral would go away leaving (sqrt(x)cos(x))/2x

Answer 3

correct

Answer 4

Which is still 0/0, so I would do it again.
Then I get -sin(x)/(2*2sqrt(x)), which is 1/0, what do I do now?

Answer 5

how did you get -sin(x) / (4sqrt(x))?

Answer 6

before that, did you get sqrt(x) cos(x) / (2x) ?

Answer 7

Yes, I think I forgot product rule on top

Answer 8

sin x can't be 0, we get cos(x) / sqrt(x) yes?

Answer 9

*since*

Answer 10

so when x->0+, cos(x) -> 1, while sqrt(x) -> 0,

so you're dividing a smaller and smaller number, which makes the fraction bigger, hence infinity

Answer 11

Is 1/0 always infinity?

Answer 12

no, 1/0 is undefined. We're talking about limit here.

Answer 13

Ok, I think I got it.
Thx