Question

Evaluate the following limit.

Answer 1

does t have any relation with x ?

Answer 2

the question is
\[\LARGE \lim_{x \rightarrow 0} \frac{\int\limits_{0}^{x} \sin(t^3)dt}{x^4}\]
according to book answer should be 1/4,

Answer 3

this is simple just apply L'Hôpital's rule

\[\Large \frac{d}{dx}(\frac{\int\limits_{0}^{x} \sin(t^3)dt}{x^3})\]
using FTC (fundamental theorem of calculus ) part1 in the numerator then
\[\Large \lim_{x \rightarrow 0} \frac{\sin(x^3)}{4x^3}=\frac{1}{4}\]

Answer 4

wow great @sami-21 nevr thought of that ..do you have a solution withou L hospital ?

Answer 5

its possible with L'Hôpital's rule becasue condition of 0/0 is satisdfied .

Answer 6

I had been working on it for quite some time without any luck in that direction

Answer 7

happens sometimes :)

Answer 8

there was a typo in the first response in the denominator it is x^4
\[\Large \frac{d}{dx} (\frac{\int\limits_{0}^{x}\sin(t^3)dt}{x^4})\]