Question

Find the limit:

Answer 1

\[(\int\limits_{0}^{x}(3t-1)^{10} dt)/3x\]

Answer 2

as x approaches 0

I'm confused because the question has two variables: x and t. How should I approach this?

Answer 3

0

Answer 4

Evaluate the limit in the top first.

Answer 5

sorry, I didn't see 3x :)

Answer 6

Use L'Hospital rule

Answer 7

Oh right. L'Hopital's rule is the best choice:
\[\frac{d}{dx}\int\limits_0^x(3t-1)^{10}dt=(3x-1)^{10}.\]

Answer 8

The derivative of the bottom is obviously 3. Hence the limit is \(\frac{3(0)-1)^{10}}{3}=\frac{1}{3}\).

Answer 9

This makes sense, but how did you know to use l'hopital's rule?

Answer 10

If we plug x=0, you will get \(\frac{0}{0}\). If you plug x in the top, you get an integral from 0 to 0, which results in a value of 0. I think it's very obvious in the denominator.

Answer 11

oh okay, thanks a ton!