Problem: Evaluate $$\lim\limits_{h \to 0}\frac{\int_{0}^{\frac{\pi}{3}+h^4e^{\frac{1}{h^2}}}\cos^{3}x\,dx -\int_{0}^{\frac\pi3}\cos^{3}x\,dx}{h^4e^{\frac{1}{h^2}}}.$$
My attempt: First, I broke the first integral in the numerator with limits from $0$ to $\frac\pi3$ and $\frac\pi3$ to the other limit. Then the two from $0$ to $\frac\pi3$ cancel out and we are left with a single integral. Now, I used L'Hospital's rule and Newton-Leibniz rule in the numerator which gives me another difficult limit.
Can someone suggest an alternative path or maybe point out a flaw in my method?
Best Answer
Hint: Note that the integrand is periodic. You can reduce this problem to finding
$$ \lim_{t\to\infty} \frac{1}{t}\int_0^t\cos^3(x)dx.$$