How to calculate $\lim\limits_{x\to1^+}\frac{1}{(x-1)^2} \int\limits_{1}^{x} \sqrt{1+\cos(\pi t)}\,\mathrm dt$ Can anyone help me by calculating this limit?
I know that I need L'Hôpital but how?
$$
\lim_{x \to 1^+}\frac{1}{(x-1)^2} \int_{1}^{x} \sqrt{1+\cos(\pi t)} \,\mathrm dt
$$
Thank you very much!!

Integrate the equivalent, valid when $t\gt1$, $t\to1$,
$$
\sqrt{1+\cos(\pi t)}=\sqrt2\,\sin\left(\frac\pi2(t-1)\right)\sim\frac\pi{\sqrt2}\,(t-1).
$$
This yields, for $x\to1$, $x\gt1$,
$$
\int_1^x\sqrt{1+\cos(\pi t)}\mathrm dt\sim\frac\pi{\sqrt2}\int_1^x(t-1)\mathrm dt=\frac\pi{2\sqrt2}(x-1)^2.
$$
