Find:
$$\lim_{n \to \infty} n \int_0^1 (\cos x – \sin x)^n dx$$
This is one of the problems i have to solve so that i could join college. I tried using integration by parts, i tried using notations but nothing works. If someone could please help me i would deeply appreciate it. ! thanks in advance ! I know the answer to the limit is 1. But i need help proving it.
Best Answer
Write the integrand as $(\cos(x)-\sin(x))^n=e^{n\log(\cos(x)-\sin(x))}$. Looking at a few plots it becomes immediatly clear the integral will be dominated by a small region around the origin with width $\epsilon\sim1/n$ as $n \rightarrow\infty$. The more formal reason for that is, that the exponent is nearly zero around the origin
$$ I_n=\int_0^1(\cos(x)-\sin(x))^n\sim \int_0^{\epsilon}e^{n\log(\cos(x)-\sin(x))} $$
Taylor expansion of the exponent yields
$$ I_n\sim\int_0^{\epsilon}e^{-x n} $$
by pushing $\epsilon$ to infinity we introduce only an exponentially small error so
$$ I_n \sim\int_0^{\infty}e^{-x n}=\frac{1}{n} $$
which yields