limits involving infinity infinite limits we have concluded that
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Limits Involving Infinity
Infinite Limits
We have concluded thatexistnot does
21
0lim xx
Limits Involving Infinity
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.
Limits Involving Infinity
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
Limits Involving Infinity
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
Limits Involving Infinity
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
Limits Involving Infinity
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
Limits Involving Infinity
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
Limits Involving Infinity
For example, the y-axis is a vertical asymptote of the curve y = 1/x2 because
21limxox
Limits Involving Infinity
Example:
Find and .
Look at the graph of this function on your graphing calculator!
32
3lim
xx
x 32
3lim
xx
x
Limits Involving Infinity
and
x
xln
0lim
x
xtan
2/lim
Limits Involving Infinity
Example:
Find .
x
x2tanln
0lim
Limits Involving Infinity
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
Limits Involving Infinity
This is expressed symbolically by writing
11212
lim
xx
x
Limits Involving Infinity
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
Limits Involving Infinity
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”
Limits Involving Infinity
Look at the illustrations on page 134, figure 9.
Limits Involving Infinity
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
Limits Involving Infinity
An example of a curve with two horizontal asymptotes
is y = tan-1 x.
21tanlim
x
x 21tanlim
x
x
Limits Involving Infinity
Find the infinite limits, limits at infinity, and asymptotes for the function f whose graph is
shown in figure 12 (page 135)
Limits Involving Infinity
Example:
Find and
We now have proven that y = 0 is a horizontal asymptote for the graph of y = 1/x.
xx1lim
xx1lim
Limits Involving Infinity
Important Rule for Calculating Limits
If n is a positive integer, then
01lim nxx
01lim nxx
Limits Involving Infinity
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
Limits Involving Infinity
Example:
Compute
xxx
12lim
Limits Involving Infinity
0lim
xex
Limits Involving Infinity
Example:
Evaluate xex
/10
lim
Limits Involving Infinity
Example:
Evaluate xx
sinlim
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