3

Part of 2017 ISI Entrance Examination

Problems(1)

Find $f'(1)$ and evaluate a limit

Source: Bstat-Bmath 2017 : Problem 3

f:\mathbb{R} \to \mathbb{R}

is a function given by f(x) =\begin{cases} 1 & \mbox{if} \ x=1 \\ e^{(x^{10}-1)}+(x-1)^2\sin\frac1{x-1} & \mbox{if} \ x\neq 1\end{cases}
(a) Find

f'(1)

\displaystyle \lim_{u\to\infty} \left[100u-u\sum_{k=1}^{100} f\left(1+\frac{k}{u}\right)\right]

functionalgebraanalytic geometrycombinatoricscalculus computations