MathDB

(a) Let

f(x)=x-xe^{-\frac1x}, \ \ x>0

. Show that

f(x)

is an increasing function on

(0,\infty)

, and

\lim_{x\to\infty} f(x)=1

.
(b) Using part (a) or otherwise, draw graphs of

y=x-1, y=x, y=x+1

, and

y=xe^{-\frac{1}{|x|}}

for

-\infty<x<\infty

using the same

X

and

Y

axes.

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