Mathematics

Problem

Consider the function

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

given by

f(x)=\begin{cases} \displaystyle\frac{e^{x}-1-x}{x^{2}}, & x\neq 0,\\ A, & x=0. \end{cases}

Answer the following. a. Part 1: (i) Show that

\displaystyle\lim_{x\to0}\frac{e^{x}-1-x}{x^{2}}

exists and find its value. b. Part 2: (ii) Hence determine the constant

A

so that

f

is continuous at

x=0

. c. Part 3: (iii) Verify that with this value of

A

, the function

f

is continuous on the whole of

\mathbb{R}