Lesson

#4 supplement------Brief summar? on Limits

& Continuity

+

Limits

1)

Ways limits fail

to

exist:

a)

the right side limit does

not equal the left

side limit

b)

the.function approaches

plus

or minus

infinity as x approaches

a

c)

the function oscillates as x approaches

a

2)

['low to evaluate

a

limit:

a) simply substitute the value that

x is approaching

in fbr

x.

b)

in the

case

where

pait

a

gives

0/0,

use the thctoring

method

c) if

parts

a or b do not

apply, use the conjugate

method

Continuity

l)

Ways a function can

be

discontinuous

at a

point

a)

if

the

right

side

limit

does not

equat the

left side

limit at that

point.

(this

is

a

jump

discontinuiff)

b) if there is a hole in the

graph

at the

point

of interest.

(this

is

a

point

discontiluityl

c)

if there is

a

vertical

asymptote

at that x value.

(this

is

an

inlinite

discontinuity)

Note:

1)

if a firnction is

continusu$

at x:a, all aspects

of

the following

limit must

hold:

..

,

,

.,,

'i

Lim

f(x)

=

f(a)

x-+a

[i.e.,

the limit must exist,

f(a) must

be defined,

& that limit

must

equal f(a)]