limits involving infinity infinite limits we have concluded that

Limits Involving Infinity

Infinite Limits

We have concluded thatexistnot does

21

0lim xx


x 1/x2

±1 1±0.5 4±0.2 25±0.1 100±0.05 400±0.01 10,000

±0.001 1,000,000

By observing from the table of values and the graph,the values of 1/x2 can be made arbitrarily large by taking x to be close enough to 0. The values of f(x) donot approach a number.


To indicate this kind of behavior, we use the notation

This does not mean that is a number. Nor does it mean that the limit exists. It expresses the particular way the limit does not exist.

2

10

limxx


In general we write to indicate that the

values of f(x) become larger and larger (or “increase without bound”) as x approaches a.

Graphical illustration on page 131, figure 2 and figure 3 and figure 4

)(lim xfax


Definition: The notation means that the

values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a (on either side of a) but not equal to a.

This notation is often read as:“the limit of f(x), as x approaches a, is infinity”“f(x) becomes infinite as x approaches a”“f(x) increases without bound as x approaches

a.

)(lim xfax


Similarly

means that values of f(x) are as large negative as we like for all values of x that are

sufficiently close to a, but not equal to a.

Look at figure 4 on page 132.

)(lim xfax


Definition: The line x = a is called a vertical asymptote

of the curve y = f(x) if at least one of the following statements is true:

)(lim xfax

)(lim xf

ax

)(lim xfax

)(lim xfax

)(lim xf

ax

)(lim xfax


For example, the y-axis is a vertical asymptote of the curve y = 1/x2 because

21limxox


Example:

Find and .

Look at the graph of this function on your graphing calculator!

32

3lim

xx

x 32

3lim

xx

x


and

x

xln

0lim

x

xtan

2/lim


Example:

Find .

x

x2tanln

0lim


Limits at Infinity

Let’s look at the graph of

As you can see from the graph, as x grows larger and larger, the values of f(x) get closer

and closer to 1.

1212

)(

xxxf


This is expressed symbolically by writing

11212

lim

xx

x


Definition:

Let f be a function defined on some interval (a, ). Then means that the values of f(x) can

be made as close to L as we like by taking x sufficiently large.

Lxf

x

)(lim


This is often read as:

“the limit of f(x), as x approaches infinity, is L”

“the limit of f(x), as x becomes infinite, is L”

“the limit of f(x), as x increases without bound, is L”


Look at the illustrations on page 134, figure 9.


Definition:

The line y = L is called a horizontal asymptote of the curve y = f(x) if either

or

Therefore the curve has a horizontal asymptote at

y = 1.

Lxfx

)(lim Lxfx

)(lim


An example of a curve with two horizontal asymptotes

is y = tan-1 x.

21tanlim

x

x 21tanlim

x

x


Find the infinite limits, limits at infinity, and asymptotes for the function f whose graph is

shown in figure 12 (page 135)


Example:

Find and

We now have proven that y = 0 is a horizontal asymptote for the graph of y = 1/x.

xx1lim

xx1lim


Important Rule for Calculating Limits

If n is a positive integer, then

01lim nxx

01lim nxx


Example:

Evaluate

To evaluate the limit at infinity of a rational function, we first divide both the numerator and denominator by the highest power of x. (We assume that x does not equal zero, since we are interested only in large values of x)

1425223lim

xx

xxx


Example:

Compute

xxx

12lim


0lim

xex


Example:

Evaluate xex

/10

lim


Example:

Evaluate xx

sinlim

limits involving infinity infinite limits we have concluded that

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