The number Lis called the limit of the function The number Lis called the limit of the function f(x) as xapproaches a, written

x

x

0x x0x x

1lim 2x

h x1x

1lim 1x

h x

lim does not existh x1

lim does not existx

h x

) If f(x) = c ( a constant function), then) If f(x) = c ( a constant function), thenfor any a

x ax a

) where n is a positiven n

) where n is a positiveintegerx a

) If and f x g x) If and x a

f xx a

g x

exist , then

x a x a x a

) If and lim ( )f x g x) If and lim ( )x a

f xx a

g x

exist , then

x a x a x a

- If exists, then for any- If exists, then for anylim ( )x a

f x

constant k,

lim ( )] lim ( )x a x a

k f x k f xx a x a

lim ( )x a

f xx a

g x

lim ( )( )lim

( ) lim ( )x a

f xf x

g x g xlim

( ) lim ( )x ax a

g x g x

) If exits, and n is a positive ) If exits, and n is a positive lim ( )x a

f x

integer

nn

x an

x a x a

IfIf

does not exist, since

) 1) 1lim 0 where p > 0px xx x

1lim 0

where p > 01

lim 0

where p > 0px x

The limit does not existThe limit does not exist

If f(x) is a rational function and is the If f(x) is a rational function and is the term with greatest power in the numerator term with greatest power in the numerator and is the term with greatest power in the denominator, thenthe denominator, then

A function f(x) is continuous at a point b if and A function f(x) is continuous at a point b if and only if:) f(x) is defined at x = b) f(x) is defined at x = b

) lim f(x) existx bx b

) ) lim f(x) = f(b)) ) lim f(x) = f(b)x b

Show that f(x) = / (x- ) is continuous at x=Show that f(x) = / (x- ) is continuous at x=and discontinuous at x = and discontinuous at x = f( )= / lim x f(x) = / =f( )

At x= the function is not defined.At x= the function is not defined.

The function is not defined at x=-So it is not continuous at x=- , but So it is not continuous at x=- , but the limit exist.the limit exist.

A polynomial function is continuous at all A polynomial function is continuous at all points of its domain. points of its domain.

Find all points of discontinuity ofFind all points of discontinuity off(x) = x - x + f(x) = x - x +

X + X -

X + X - = (x+ ) (x- ).

The denominator is zero when x=The denominator is zero when x=-- or x=or x=The denominator is zero when x=The denominator is zero when x=-- or x=or x=Thus the function is discontinuous at x=-Thus the function is discontinuous at x=-and x= only.