Finding $f'(0)$ when $f(x)=\int\limits_0^x\sin\left(\frac{1}{t}\right)dt$

I need to show that $f'(0)=0$ for $$ f(x)=\int\limits_0^x\sin\left(\frac{1}{t}\right)dt $$ But fundamental theorem of calculus is unapplicable here. What should I do?

Solution 1:

Here is one approach:

• Use integration by parts to show that $f(x)=x^2\cos(1/x)-\int_0^x 2t\cos(1/t)dt$ for $x\neq 0$.
• Use this to show that $\left|\dfrac{f(x)}{x}\right|\leq 2|x|$.

Solution 2:

To strengthen the convergence of an integral, a integration by parts is always a good idea. Here we have $$\begin{aligned} f(x) &=\int_0^x\sin\left(\frac 1t \right) dt\\ &= \left[ t^2 \cos\left(\frac 1t \right)\right]_0^x - \int_0^x 2t\cos\left(\frac 1t \right)dt\\ &= x^2 \cos\left(\frac 1x \right) - \int_0^x 2t\cos\left(\frac 1t \right)dt \end{aligned}$$ You can apply the fundamental theorem of calculus for the second term, since it is the integral of a continuous function. The first term is a $O(x^2)$, so it is differentiable at zero with null derivative. In the end, we get $f'(0) = 0$.

[Reminder — A function $f$ is derivable at zero with derivative $a$ if and only if $f(x) = f(0) + ax + o(x)$ when $x\to 0$.]