In previous sections we have been using calculators and graphs to guess the values of limits. Sometimes, these methods do not work! In this section we.

In previous sections we have been using calculators and graphs to guess the values

of limits.

Sometimes, these methods do not work!

In this section we will use properties of limits, called limit laws, to calculate limits.

Calculating Limits Using the Limit Laws

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

and

exist. Then

1.

The limit of a sum is the sum of the limits.


)(lim xfax

)(lim xgax

)(lim)(lim )()(lim xgax

xfax

xgxfax

Limit Laws:

2.

The limit of a difference is the difference of the limits.


)(lim- )(lim )()(lim xgax

xfax

xgxfax

Limit Laws:

3.

The limit of a constant times a function is the constant times the limit of the function.


)(lim )(lim xfax

cxcfax

Evaluate the limit and justify each step.

Example 1:

Note: If we let f(x)=2x2-3x+4, then f(5)=39.


)4 3 - 22(5

lim

xxx

Limit Laws:

4.

The limit of a product is the product of the limits.


)()(lim )()(lim xgxfax

xgxfax

Limit Laws:

5.

The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0).


0 )(lim if )(lim

)(lim

)()(lim

xgaxxg

ax

xfax

xgxf

ax

Evaluate the limit and justify your answer.

Example 2:

Note: As in example 1, if we let f(x)=(x3+2x2-1)/(5-3x), thenf(-2)=-1/11


xxx

x 351223

2lim

Direct Substitution Property:

If f is a polynomial or a rational function and a is in the domain of f, then


)()(lim afxfax

Let’s try example 1 on page 111.


Example 4: Find


112

1lim

xx

x

Limit Laws cont:

6. where n is a positive integer


nxf

ax

nxf

ax

)(lim )(lim

Special Limits:

7.

8.


ccax

lim

axax

lim

More Special Limits:

9. where n is a positive integer

10.


nanxax

lim

]0a that assume weeven, isn [if

integer positive a isn where lim

n an xax

More Limit Laws:

11.

where n is a positive integer[If n is even, we assume that ]


n xfax

n xfax

)(lim lim

0 )(lim

xfax

Example 5:

Find

Remember: The limit is not necessarily what the function is equal to; it is what the function approaches when x gets close to a


1 xif 1 xif 1)( where)(

1lim

xxgxg

x

Example 6:

Evaluate:

Sometimes we may have to use algebra to take limits.


hh

h

92)3(

0lim

Example 7:

Find

Here we have to rationalize the numerator in order to find the limit.


2392

0lim

t

t

t

Theorem:

Some limits are best calculated by first finding the left- and right- hand limits. This theorem is a reminder of what we discovered in previous sections.


)(lim)(-

limifonly and if )(lim xfax

Lxfax

Lxfax

Example 8:

Show that


00

lim

xx

Example 9:

Prove that does not exist.


x

x

x 0lim

Example 10:

Show that does not exist.


xx 3lim

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 xgax

xfax

The Squeeze Theorem:

If f(x) < g(x) < h(x) when x is near a (except possibly at a) and

then


Lxhax

xfax

)(lim)(lim

Lxgax

)(lim

Example 11:

By using the Squeeze Theorem show that


01sin20

lim x

xx

In previous sections we have been using calculators and graphs to guess the values of limits. Sometimes, these methods do not work! In this section we.

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