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ISI B.Stat Entrance Exam
2006 ISI B.Stat Entrance Exam
6
f(x)=x-xe^(-1/x)
f(x)=x-xe^(-1/x)
Source: ISI (BS) 2006 #6
June 2, 2012
function
limit
calculus
calculus computations
Problem Statement
(a) Let
f
(
x
)
=
x
−
x
e
−
1
x
,
x
>
0
f(x)=x-xe^{-\frac1x}, \ \ x>0
f
(
x
)
=
x
−
x
e
−
x
1
,
x
>
0
. Show that
f
(
x
)
f(x)
f
(
x
)
is an increasing function on
(
0
,
∞
)
(0,\infty)
(
0
,
∞
)
, and
lim
x
→
∞
f
(
x
)
=
1
\lim_{x\to\infty} f(x)=1
lim
x
→
∞
f
(
x
)
=
1
.(b) Using part (a) or otherwise, draw graphs of
y
=
x
−
1
,
y
=
x
,
y
=
x
+
1
y=x-1, y=x, y=x+1
y
=
x
−
1
,
y
=
x
,
y
=
x
+
1
, and
y
=
x
e
−
1
∣
x
∣
y=xe^{-\frac{1}{|x|}}
y
=
x
e
−
∣
x
∣
1
for
−
∞
<
x
<
∞
-\infty<x<\infty
−
∞
<
x
<
∞
using the same
X
X
X
and
Y
Y
Y
axes.
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