Section 1.6

Calculating Limits Using the Limit Laws

Theorem:

• All the theorem is saying is that the limit has to be the same from left and right for the limit to exist.

• Direct substitution property: If f is a polynomial or a rational function and a is in the domain of f, then

• In other words as long as we don not have 0/0, then we can evaluate the limit by substitution.

)(lim)(lim)(lim xfLxfifonlyandifLxfaxaxax

)()(lim afxfax

Limit Laws: Suppose that c is a constant and the limits

• 1

• 2

• 3

• 4

Thenexistxgandxf axax .)(lim)(lim

)(lim)(lim)()(lim xgxfxgxf axaxax

)(lim)(lim xfcxcf axax

)(lim)(lim)()(lim xgxfxgxf axaxax

0)(lim)(lim

)(lim

)(

)(lim

xgif

xg

xf

xg

xfax

ax

axax

5. naxn

ax xfxf )(lim)(lim

axcc axax lim.7lim.6

egerpositiveaisnwhereax nnax intlim.8

0)(,

)(lim)(lim.11

0,

intlim.10

xfthatassumeweevenisnIf

xfxf

athatassumeweevenisnIf

egerpositiveaisnwhereax

nax

nax

nnax

Examples

Evaluate the following limits:

1

1lim

35

12lim

432lim

2

1

23

2

25

x

x

x

xx

xx

x

x

x

More Examples

• Evaluate

x

x

x

t

t

h

h

x

x

t

h

0

0

2

2

0

2

0

lim

lim

39lim

93lim

Theorem: If f(x) g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a. Then:

)(lim)(lim xgxf axax

The Squeeze Theorem

Lxg

Then

Lxhxf

andaatpossiblyexceptanearisxwhenxhxgxfIf

ax

axax

)(lim

)(lim)(lim

)()()()(

Example of The Squeeze Theorem

• Show that the 0/1sinlim 20 xxx