Help evaluating $\lim\limits_{h\to0}\int_{\frac{\pi}{4}}^{\frac{\pi}{4}+h}\frac{\sin(x)}{x}dx$ (multiple choice) 
Evaluate
  $\lim\limits_{h\to0}\int_{\frac{\pi}{4}}^{\frac{\pi}{4}+h}\frac{\sin(x)}{x}dx$
A. $0$
B. $\dfrac{\sqrt{2}}{2}$
C. $1$
D. $\dfrac{2\sqrt{2}}{\pi}$
E. $\dfrac{\sqrt{2}}{2\pi}$

I was under the assumption that I could plug in $0$ for $h$ and conclude that the integral is zero, making my answer choice A. However, the answer key states that D is the correct answer. 
I noticed that you could arrive at D by evaluating the integrand at $x=\dfrac{\pi}{4}$, but I'm not sure why that's justified. Is the key wrong, or is there a deeper layer of understanding to this question?
Any help is appreciated!

$$L= \lim_{h \rightarrow 0} \int_{\pi/4}^{\pi/4+h} \frac{\sin x}{x} dx =0$$ As the lower anmd upper  limits of the integral coincide and $f(x)=\frac{\sin x}{x}$ is finte around (in the neighbour of) $x=\pi/4$. 
