calculate $ \lim \limits_{x \to 0} x^a \int_x^1 \frac{f(t)}{t^{a+1}} \, dt$
Correct way to prove the limit: $\mathop {\lim }\limits_{x \to 0} {x^\alpha }\int\limits_x^1 {\frac{{f\left( t \right)}}{{{t^{\alpha  + 1}}}}dt} $?