## anonymous 4 years ago Find the limit:

1. anonymous

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

2. anonymous

as x approaches 0 I'm confused because the question has two variables: x and t. How should I approach this?

3. watchmath

0

4. Mr.Math

Evaluate the limit in the top first.

5. watchmath

sorry, I didn't see 3x :)

6. watchmath

Use L'Hospital rule

7. Mr.Math

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

8. Mr.Math

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

9. anonymous

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

10. Mr.Math

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.

11. anonymous

oh okay, thanks a ton!