Example Ex. Find all asymptotes of the curve Sol. So x=3 and x=-1 are vertical asymptotes. So y=x+2...

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