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Question Number 101008 by Rio Michael last updated on 29/Jun/20

Given f (x) = {\begin{cases} \frac{1}{2} {x e}^{\frac{1}{x}}, x \neq 0 \\ 0, x = 0 \end{cases}

find

(i) Thd domain of f

(i i) check the continuity of f at x = 0

(i i i) check its differentiability and its sign

(i) sketch this curve and find \lim_{x \to - \infty} f (x) and \lim_{x \to + \infty} f (x)

Answered by maths mind last updated on 29/Jun/20

D_{f} = R

2) \lim_{x \to 0^{-}} f (x) = 0

3) \lim_{x \to 0^{+}} f (x) = + \infty f n o t c o n t i n u e

(i i i) d i f f r e n t i a b i l i t e \Rightarrow c o n t i n u i t y

\Leftrightarrow (n o c o n g i n u i t y \Rightarrow n o d i f f e r e n t i a b i l i t i e)

\lim_{x \to - \infty} f (x) = - \infty, \lim_{x \to + \infty} f (x) = + \infty