(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] A limit problem

I just need some help showing how this limit systematically follows from the limit rules:

My first chain of thought led to breaking the expression up as follows:Code (Text):

# x 2

1 # t 1

lim --- # ---------- dt = ---

x->0 3 # 2 3

x # 0 t + 1

1/x^2 * ( 1/x * Integral[0..x, t^2/(t^2+1)] )

Then I just kind of figured that the subexpression on the right was the average value of the function being integrated from 0..x and as x->0 the average value would approach x^2/(x^2+1), which led to:

1/x^2 * x^2/(x^2+1) = 1/(x^2+1)

which would approach one as x approached zero.

But clearly that's wrong (not surprisingly since I made a sketchy move in the middle) since the answer is one-third. Can anyone show me how to do this?

edit: possibly w/o actually integrating because this is an exercise in which you're expected to know the FTofC but not how to integrate that.

edit2: oh, not supposed to no l'hospital's rule either.

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# A limit problem

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