The right-hand limit of f (x), as x approaches a, equals L

written:

if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a.

lim ( )x a

f x L

a

L( )y f x

One-Sided Limits

The left-hand limit of f (x), as x approaches a, equals M

written:

if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a.

lim ( )x a

f x M

a

M

( )y f x

lim ( ) if and only if lim ( ) and lim ( ) .x a x a x a

f x L f x L f x L

Theorem

Find2

3 if 2lim ( ) where ( )

1 if 2x

x xf x f x

x

-2

6

Note: f (-2) = 1

Examples

1.

2.

3.

4.

2.6 Continuous and discontinuous functions A function f is continuous at a point x = a if the following are true:

) ( ) is definedi f a) lim ( ) exists

x aii f x

a

f(a)) lim ( ) ( )

x aiii f x f a