For what values of p q does limt rightarrow 0 integral1 t si

For what values of p, q does lim_t rightarrow 0+ integral^1 _t sin (x^p)/x^q dx converge? Explain.

Solution

solution:For p=q=1 above integration would be finite.

also

This is valid if x is not equals to zero because division by zero is not allowed. So, in case of definite integration,

and in question it is given already that lim is tending to 0+

and above series is convergent in interval [0+ ,1]