Thomas Calculus 13th [Solutions]

Section 4.5 Indeterminate Forms and LHôpitals Rule 287
(b) The limit leads to the indeterminate form :
2 2
2
2 2
x x x
x x x
lim x x x lim x x x
lim lim x
2 2 2
x x x x x x x x x x x x x
lim 1 1 1
x 1 1 1 1 0 2
1
x
82.
2 2
2 1
lim x 1 x lim x x x lim x x 1 x lim x 1 1 1
x x 2 2 2
x x x x x x
x x
83. The graph indicates a limit near 1. The limit leads
2
2 (3 1) 2
to the indeterminate form 0 : lim x x x
0
x 1
x 1
9 1/2 1 1/2
lim 2 3/2 1/2
2x 3x x 2
4
2 2
1
1 lim x x x
x
x
x 1
1
4
9 1
2 2 4 5 1
1 1
84. (a) The limit leads to the indeterminate form 1 . Let f ( x x
) 1 1 ln ( ) ln 1 1 lim ln ( )
x
f x x x
f x
x
2
1
1
x
ln 1 x
x
1 x
1
1 1 2
x
x
1
1
x
x
lim ln 1
lim lim lim 1 1 1
1 0
1 lim 1 x
lim f
x
( x )
x x x x x x
ln f ( x) lim e
1
e e
x
(b) x 1 1
x
x
10 2.5937424601
100 2.70481382942
1000 2.71692393224
10,000 2.71814592683
100,000 2.71826823717
Both functions have limits as x approaches
infinity. The function f has a maximum but no
minimum while g has no extrema. The limit of
f ( x ) leads to the indeterminate form 1 .
x
2
(c) Let f ( x) 1 1 ln f ( x) x ln 1 x
2
x
2
3
2
x
x
1 x
2
2
x
1 2 3 2
lim ln
ln 1
f ( x ) lim lim lim 2 lim 4x
lim 4
3 1 6x
0.
x x x x x x x x x x x
x
ln ( ) 0
Therefore lim 1 1
f x
lim f ( x) lim e e 1
2
x x x x
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