V

U

qi{s

iry"p.!xie*

inlinity

lim

f(xll- (limit

exist)

(horizontal

asymptote).

show

graphicalty.

x-)m

Iim

f(x)=L

(similar

to abow)

x-+-00

Iim

f(x)=1oo (4

cases)

(no

limit

for

each

case).

Show

graphically.

x-+t6

-Other

examples:

lim (2x2

-x+5),

h

[(*'

-fi/(x-z)f,

tm

[1:x*

*z*,

-sy(z*o

-5*,

+6]

(glve

rule)

x-+1

x*+2

E-+m

I*

k\ff,

-z\(x-+l

1-sd

Formal

definition

of

e

limit

-You

are not

responsible

for

this.

see cauchy,s

definitiotr

on my

website,

-To

prove

lim f(xll,'

one

must

show

the

following:

For

any

positive

number,

e,

there

x-)a

exists

a

positive

number

6,

such

that if

[x-af<6,

then

l(x)-Lt<e

*symbolic

form:

Vp0,

lE>O,

a

[x-al<6

-+

lf(x)-Ll<e

-Thffe

are

similar

definitions

for

other

fypes

of limits

(involving

infinity)

-see

my

website

for

a simple proof

using

cauchy,s

definition

of a

timit.