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Example Ex. Find all asymptotes of the curve
Sol.
So x=3 and x=-1 are vertical asymptotes.
So y=x+2 is a slant asymptote.
3
2.
2 3
x
yx x
2
3 12 3 ( 3)( 1) lim lim
x xx x x x y y
lim / 1, lim( ) lim( ) 2
x x x
m y x b y mx y x
Example Ex. Find asymptotes of the curve Sol. vertical asymptote
horizontal asymptote
Ex. Find asymptotes of the curve Sol. vertical asymptote
slant asymptote
| |( ) .
1
x
f xx
2( 2).
2( 1)
x
yx
1x1, 1 y y
1x
3
2 2 x
y
Curve sketching A. Domain B. Intercepts C. Symmetry D. Asymptotes E. Intervals of increase or decrease F. Local maximum and minimum values G. Convexity and points of inflection H. Sketch the curve
Example Ex. Sketch the graph of Sol. A. The domain is (-1,+1). B. The y-intercept is 1.
C. f is even. D. asymptotes: y=0 is horizontal asymptote.
E. when x>0, so f(x) decreasing
in (0,+1) and increasing in (-1,0).
F. x=0 is local and global maximum point.
G. f(x) concave in and
convex otherwise
2
. xy e
2
( ) 2 , xf x xe
2 2( ) 2 (2 1), xf x e x 2 2( , )
2 2
( ) 0, f x
Example Sketch the graph of
2
2
2.
(1 )
x
yx
Example Ex. Prove the inequality:
Proof. Let then
2 2( 1) ln ( 1) ( 0).x x x x
1( ) 2 ln 2f x x x x
x
2
1( ) 2 ln 1f x x
x
2 2( ) ( 1) ln ( 1) ,f x x x x
2
3
2( 1)( )
xf x
x
Indeterminate forms Question: find the limit we can’t apply the limit
law because the limit of the denominator is 0. In fact the limit
of the numerator is also 0. We call this type of limit an
indeterminate form.
Generally, if both and as
then the limit
may or may not exist and is called an indeterminate form of
type 0/0
1
lnlim
1x
x
x
( ) 0f x ( ) 0g x ,x a
( )lim
( )x a
f x
g x
Previous methods For rational functions, for example,
The important limit:
Does not work for general cases. There is a systematic
method, known as L’Hospital’s Rule, for evaluation of
indeterminate forms.
2
21 1 1
( 1) 1lim lim lim .
1 ( 1)( 1) 1 2x x x
x x x x x
x x x x
0
sinlim 1x
x
x
L’Hospital’s ruleL’Hospital’s Rule Suppose f and g are differentiable and
near a (except possibly at a). Assume that
and
or that
and
Then
if the last limit exists (can be a real number or or ).
( ) 0 g x
lim ( ) 0
x a
f x lim ( ) 0
x ag x
lim ( )x a
f x
lim ( )x ag x
( ) ( )lim lim
( ) ( )
x a x a
f x f x
g x g x
Remarks Remark1. L’Hospital’s Rule can be used to evaluate the
indefinite limit of type 0/0 or 1/1.
Remark2. L’Hospital’s Rule is also valid when “x!a” is
replaced by x!a+, x!a-, x!+1, x!-1.
Remark3. If is still an indeterminate type, we can
use L’Hospital’s Rule again.
( )lim
( )
x a
f x
g x
ExamplesEx. Find
Sol.
Ex. Find
Sol.
Note: This example indicates that exponential infinity is
much bigger than any power infinity.
30
sinlim .
x
x x
x
3 20 0 0
sin 1 cos sin 1lim lim lim .
3 6 6x x x
x x x x
x x x
lim ( 1).
n
xx
xa
a1 !
lim lim lim 0.ln (ln )
n n
x x x nx x x
x nx n
a a a a a
When not to use L’Hospital’s rule
Ex. Find
Sol.
L’Hospital’s Rule gives nothing! Correct solution is 0.
2
0
1sin
lim .sinx
xxx
2
0 0
1 1 1sin 2 sin cos
lim limsin cos
x x
x xx x xx x
Other indeterminate types There are some other indeterminate types which can be
changed into 0/0 or 1/1 type: 0¢1, 1-1, 11, 10, 00.
Ex. Find
Sol.
0lim ln . xx x
0 0 0
2
1ln
lim ln lim lim 0.1 1
x x x
x xx x
x x
Homework 10 Section 4.4: 22, 30, 31, 45, 48, 55, 74
Section 4.5: 26, 28, 64
