Transcendental functions such as exponential, trigonometric and hyperbolic functions (Part - 3) | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


4.10 Limits revisited We have de?ned and used the concept of limit, primarily in our development of the deriva-
tive. Recall that lim
x?a
f(x) = L is true if, in a precise sense, f(x) gets closer and closer to
L as x gets closer and closer to a. While some limits are easy to see, others take some
ingenuity; in particular, the limits that de?ne derivatives are always di?cult on their face,
since in
lim
?x?0
f(x+?x)-f(x)
?x
both the numerator and denominator approach zero. Typically this di?culty can be re-
solved when f isa“nice” function and we are tryingtocompute a derivative. Occasionally
such limits are interesting for other reasons, and the limit of a fraction in which both nu-
merator and denominator approach zero can be di?cult to analyze. Now that we have
the derivative available, there is another technique that can sometimes be helpful in such
circumstances.
Before we introduce the technique, we will also expand our concept of limit, in two
ways. When the limit of f(x) as x approaches a does not exist, it may be useful to note in
what way it does not exist. We have already talked about one such case: one-sided limits.
Another case is when “f goes to in?nity”. We also will occasionally want to know what
happens to f when x “goes to in?nity”.
EXAMPLE 4.10.1 What happens to 1/x as x goes to 0? From the right, 1/x gets
bigger and bigger, or goes to in?nity. From the left it goes to negative in?nity.
EXAMPLE 4.10.2 What happens to the function cos(1/x) as x goes to in?nity? It
seems clear that as x gets larger and larger, 1/x gets closer and closer to zero, so cos(1/x)
should be getting closer and closer to cos(0) = 1.
Page 2


4.10 Limits revisited We have de?ned and used the concept of limit, primarily in our development of the deriva-
tive. Recall that lim
x?a
f(x) = L is true if, in a precise sense, f(x) gets closer and closer to
L as x gets closer and closer to a. While some limits are easy to see, others take some
ingenuity; in particular, the limits that de?ne derivatives are always di?cult on their face,
since in
lim
?x?0
f(x+?x)-f(x)
?x
both the numerator and denominator approach zero. Typically this di?culty can be re-
solved when f isa“nice” function and we are tryingtocompute a derivative. Occasionally
such limits are interesting for other reasons, and the limit of a fraction in which both nu-
merator and denominator approach zero can be di?cult to analyze. Now that we have
the derivative available, there is another technique that can sometimes be helpful in such
circumstances.
Before we introduce the technique, we will also expand our concept of limit, in two
ways. When the limit of f(x) as x approaches a does not exist, it may be useful to note in
what way it does not exist. We have already talked about one such case: one-sided limits.
Another case is when “f goes to in?nity”. We also will occasionally want to know what
happens to f when x “goes to in?nity”.
EXAMPLE 4.10.1 What happens to 1/x as x goes to 0? From the right, 1/x gets
bigger and bigger, or goes to in?nity. From the left it goes to negative in?nity.
EXAMPLE 4.10.2 What happens to the function cos(1/x) as x goes to in?nity? It
seems clear that as x gets larger and larger, 1/x gets closer and closer to zero, so cos(1/x)
should be getting closer and closer to cos(0) = 1.
As with ordinary limits, these concepts can be made precise. Roughly, we want
lim
x?a
f(x) =8 to mean that we can make f(x) arbitrarily large by making x close enough
to a, and lim
x?8
f(x) = L should mean we can make f(x) as close as we want to L by
making x large enough. Compare this de?nition to the de?nition of limit in section 2.3,
de?nition 2.3.2.
DEFINITION 4.10.3 If f is a function, we say that lim
x?a
f(x) =8 if for every N > 0
there is a d > 0 such that whenever |x-a| < d, f(x) > N. We can extend this in the
obvious ways to de?ne lim
x?a
f(x)=-8, lim
x?a
-
f(x)=±8, and lim
x?a
+
f(x)=±8.
DEFINITION4.10.4 Limitatin?nity Iff isafunction,wesaythat lim
x?8
f(x) = L
if for every o > 0 there is an N > 0 so that whenever x > N, |f(x)-L| < o. We may
similarly de?ne lim
x?-8
f(x) = L, and using the idea of the previous de?nition, we may
de?ne lim
x?±8
f(x)=±8.
We include these de?nitions for completeness, but we will not explore them in detail.
Su?ce it to say that such limits behave in much the same way that ordinary limits do; in
particular there are some analogs of theorem 2.3.6.
Now consider this limit:
lim
x?p
x
2
-p
2
sinx
.
As x approaches p, both the numerator and denominator approach zero, so it is not
obvious what, if anything, the quotient approaches. We can often compute such limits by
application of the following theorem.
THEOREM 4.10.5 L’Hˆ opital’s Rule For “su?ciently nice” functions f(x) and
g(x), if lim
x?a
f(x) = 0 = lim
x?a
g(x) or both lim
x?a
f(x) =±8 and lim
x?a
g(x) =±8, and if
lim
x?a
f
'
(x)
g
'
(x)
exists, then lim
x?a
f(x)
g(x)
= lim
x?a
f
'
(x)
g
'
(x)
. This remains true if “x?a” is replaced by
“x?8” or “x?-8”.
This theorem is somewhat di?cult to prove, in part because it incorporates so many
di?erent possibilities, so we will not prove it here. We also will not need to worry about
the precise de?nition of “su?ciently nice”, as the functions we encounter will be suitable.
EXAMPLE 4.10.6 Compute lim
x?p
x
2
-p
2
sinx
in two ways.
Page 3


4.10 Limits revisited We have de?ned and used the concept of limit, primarily in our development of the deriva-
tive. Recall that lim
x?a
f(x) = L is true if, in a precise sense, f(x) gets closer and closer to
L as x gets closer and closer to a. While some limits are easy to see, others take some
ingenuity; in particular, the limits that de?ne derivatives are always di?cult on their face,
since in
lim
?x?0
f(x+?x)-f(x)
?x
both the numerator and denominator approach zero. Typically this di?culty can be re-
solved when f isa“nice” function and we are tryingtocompute a derivative. Occasionally
such limits are interesting for other reasons, and the limit of a fraction in which both nu-
merator and denominator approach zero can be di?cult to analyze. Now that we have
the derivative available, there is another technique that can sometimes be helpful in such
circumstances.
Before we introduce the technique, we will also expand our concept of limit, in two
ways. When the limit of f(x) as x approaches a does not exist, it may be useful to note in
what way it does not exist. We have already talked about one such case: one-sided limits.
Another case is when “f goes to in?nity”. We also will occasionally want to know what
happens to f when x “goes to in?nity”.
EXAMPLE 4.10.1 What happens to 1/x as x goes to 0? From the right, 1/x gets
bigger and bigger, or goes to in?nity. From the left it goes to negative in?nity.
EXAMPLE 4.10.2 What happens to the function cos(1/x) as x goes to in?nity? It
seems clear that as x gets larger and larger, 1/x gets closer and closer to zero, so cos(1/x)
should be getting closer and closer to cos(0) = 1.
As with ordinary limits, these concepts can be made precise. Roughly, we want
lim
x?a
f(x) =8 to mean that we can make f(x) arbitrarily large by making x close enough
to a, and lim
x?8
f(x) = L should mean we can make f(x) as close as we want to L by
making x large enough. Compare this de?nition to the de?nition of limit in section 2.3,
de?nition 2.3.2.
DEFINITION 4.10.3 If f is a function, we say that lim
x?a
f(x) =8 if for every N > 0
there is a d > 0 such that whenever |x-a| < d, f(x) > N. We can extend this in the
obvious ways to de?ne lim
x?a
f(x)=-8, lim
x?a
-
f(x)=±8, and lim
x?a
+
f(x)=±8.
DEFINITION4.10.4 Limitatin?nity Iff isafunction,wesaythat lim
x?8
f(x) = L
if for every o > 0 there is an N > 0 so that whenever x > N, |f(x)-L| < o. We may
similarly de?ne lim
x?-8
f(x) = L, and using the idea of the previous de?nition, we may
de?ne lim
x?±8
f(x)=±8.
We include these de?nitions for completeness, but we will not explore them in detail.
Su?ce it to say that such limits behave in much the same way that ordinary limits do; in
particular there are some analogs of theorem 2.3.6.
Now consider this limit:
lim
x?p
x
2
-p
2
sinx
.
As x approaches p, both the numerator and denominator approach zero, so it is not
obvious what, if anything, the quotient approaches. We can often compute such limits by
application of the following theorem.
THEOREM 4.10.5 L’Hˆ opital’s Rule For “su?ciently nice” functions f(x) and
g(x), if lim
x?a
f(x) = 0 = lim
x?a
g(x) or both lim
x?a
f(x) =±8 and lim
x?a
g(x) =±8, and if
lim
x?a
f
'
(x)
g
'
(x)
exists, then lim
x?a
f(x)
g(x)
= lim
x?a
f
'
(x)
g
'
(x)
. This remains true if “x?a” is replaced by
“x?8” or “x?-8”.
This theorem is somewhat di?cult to prove, in part because it incorporates so many
di?erent possibilities, so we will not prove it here. We also will not need to worry about
the precise de?nition of “su?ciently nice”, as the functions we encounter will be suitable.
EXAMPLE 4.10.6 Compute lim
x?p
x
2
-p
2
sinx
in two ways.
First we use L’Hˆ opital’s Rule: Since the numerator and denominator both approach
zero,
lim
x?p
x
2
-p
2
sinx
= lim
x?p
2x
cosx
,
provided the latter exists. But in fact this is an easy limit, since the denominator now
approaches-1, so
lim
x?p
x
2
-p
2
sinx
=
2p
-1
=-2p.
We don’t really need L’Hˆ opital’s Rule to do this limit. Rewrite it as
lim
x?p
(x+p)
x-p
sinx
and note that
lim
x?p
x-p
sinx
= lim
x?p
x-p
-sin(x-p)
= lim
x?0
-
x
sinx
since x-p approaches zero as x approaches p. Now
lim
x?p
(x+p)
x-p
sinx
= lim
x?p
(x+p) lim
x?0
-
x
sinx
= 2p(-1) =-2p
as before.
EXAMPLE 4.10.7 Compute lim
x?8
2x
2
-3x+7
x
2
+47x+1
in two ways.
As x goes to in?nity both the numerator and denominator go to in?nity, so we may
apply L’Hˆ opital’s Rule:
lim
x?8
2x
2
-3x+7
x
2
+47x+1
= lim
x?8
4x-3
2x+47
.
In the second quotient, it is still the case that the numerator and denominator both go to
in?nity, so we are allowed to use L’Hˆ opital’s Rule again:
lim
x?8
4x-3
2x+47
= lim
x?8
4
2
= 2.
So the original limit is 2 as well.
Again, we don’t really need L’Hˆ opital’s Rule, and in fact a more elementary approach
is easier—we divide the numerator and denominator by x
2
:
lim
x?8
2x
2
-3x+7
x
2
+47x+1
= lim
x?8
2x
2
-3x+7
x
2
+47x+1
1
x
2
1
x
2
= lim
x?8
2-
3
x
+
7
x
2
1+
47
x
+
1
x
2
.
Now as x approaches in?nity, all the quotients with some power of x in the denominator
approach zero, leaving 2 in the numerator and 1 in the denominator, so the limit again is
2.
Page 4


4.10 Limits revisited We have de?ned and used the concept of limit, primarily in our development of the deriva-
tive. Recall that lim
x?a
f(x) = L is true if, in a precise sense, f(x) gets closer and closer to
L as x gets closer and closer to a. While some limits are easy to see, others take some
ingenuity; in particular, the limits that de?ne derivatives are always di?cult on their face,
since in
lim
?x?0
f(x+?x)-f(x)
?x
both the numerator and denominator approach zero. Typically this di?culty can be re-
solved when f isa“nice” function and we are tryingtocompute a derivative. Occasionally
such limits are interesting for other reasons, and the limit of a fraction in which both nu-
merator and denominator approach zero can be di?cult to analyze. Now that we have
the derivative available, there is another technique that can sometimes be helpful in such
circumstances.
Before we introduce the technique, we will also expand our concept of limit, in two
ways. When the limit of f(x) as x approaches a does not exist, it may be useful to note in
what way it does not exist. We have already talked about one such case: one-sided limits.
Another case is when “f goes to in?nity”. We also will occasionally want to know what
happens to f when x “goes to in?nity”.
EXAMPLE 4.10.1 What happens to 1/x as x goes to 0? From the right, 1/x gets
bigger and bigger, or goes to in?nity. From the left it goes to negative in?nity.
EXAMPLE 4.10.2 What happens to the function cos(1/x) as x goes to in?nity? It
seems clear that as x gets larger and larger, 1/x gets closer and closer to zero, so cos(1/x)
should be getting closer and closer to cos(0) = 1.
As with ordinary limits, these concepts can be made precise. Roughly, we want
lim
x?a
f(x) =8 to mean that we can make f(x) arbitrarily large by making x close enough
to a, and lim
x?8
f(x) = L should mean we can make f(x) as close as we want to L by
making x large enough. Compare this de?nition to the de?nition of limit in section 2.3,
de?nition 2.3.2.
DEFINITION 4.10.3 If f is a function, we say that lim
x?a
f(x) =8 if for every N > 0
there is a d > 0 such that whenever |x-a| < d, f(x) > N. We can extend this in the
obvious ways to de?ne lim
x?a
f(x)=-8, lim
x?a
-
f(x)=±8, and lim
x?a
+
f(x)=±8.
DEFINITION4.10.4 Limitatin?nity Iff isafunction,wesaythat lim
x?8
f(x) = L
if for every o > 0 there is an N > 0 so that whenever x > N, |f(x)-L| < o. We may
similarly de?ne lim
x?-8
f(x) = L, and using the idea of the previous de?nition, we may
de?ne lim
x?±8
f(x)=±8.
We include these de?nitions for completeness, but we will not explore them in detail.
Su?ce it to say that such limits behave in much the same way that ordinary limits do; in
particular there are some analogs of theorem 2.3.6.
Now consider this limit:
lim
x?p
x
2
-p
2
sinx
.
As x approaches p, both the numerator and denominator approach zero, so it is not
obvious what, if anything, the quotient approaches. We can often compute such limits by
application of the following theorem.
THEOREM 4.10.5 L’Hˆ opital’s Rule For “su?ciently nice” functions f(x) and
g(x), if lim
x?a
f(x) = 0 = lim
x?a
g(x) or both lim
x?a
f(x) =±8 and lim
x?a
g(x) =±8, and if
lim
x?a
f
'
(x)
g
'
(x)
exists, then lim
x?a
f(x)
g(x)
= lim
x?a
f
'
(x)
g
'
(x)
. This remains true if “x?a” is replaced by
“x?8” or “x?-8”.
This theorem is somewhat di?cult to prove, in part because it incorporates so many
di?erent possibilities, so we will not prove it here. We also will not need to worry about
the precise de?nition of “su?ciently nice”, as the functions we encounter will be suitable.
EXAMPLE 4.10.6 Compute lim
x?p
x
2
-p
2
sinx
in two ways.
First we use L’Hˆ opital’s Rule: Since the numerator and denominator both approach
zero,
lim
x?p
x
2
-p
2
sinx
= lim
x?p
2x
cosx
,
provided the latter exists. But in fact this is an easy limit, since the denominator now
approaches-1, so
lim
x?p
x
2
-p
2
sinx
=
2p
-1
=-2p.
We don’t really need L’Hˆ opital’s Rule to do this limit. Rewrite it as
lim
x?p
(x+p)
x-p
sinx
and note that
lim
x?p
x-p
sinx
= lim
x?p
x-p
-sin(x-p)
= lim
x?0
-
x
sinx
since x-p approaches zero as x approaches p. Now
lim
x?p
(x+p)
x-p
sinx
= lim
x?p
(x+p) lim
x?0
-
x
sinx
= 2p(-1) =-2p
as before.
EXAMPLE 4.10.7 Compute lim
x?8
2x
2
-3x+7
x
2
+47x+1
in two ways.
As x goes to in?nity both the numerator and denominator go to in?nity, so we may
apply L’Hˆ opital’s Rule:
lim
x?8
2x
2
-3x+7
x
2
+47x+1
= lim
x?8
4x-3
2x+47
.
In the second quotient, it is still the case that the numerator and denominator both go to
in?nity, so we are allowed to use L’Hˆ opital’s Rule again:
lim
x?8
4x-3
2x+47
= lim
x?8
4
2
= 2.
So the original limit is 2 as well.
Again, we don’t really need L’Hˆ opital’s Rule, and in fact a more elementary approach
is easier—we divide the numerator and denominator by x
2
:
lim
x?8
2x
2
-3x+7
x
2
+47x+1
= lim
x?8
2x
2
-3x+7
x
2
+47x+1
1
x
2
1
x
2
= lim
x?8
2-
3
x
+
7
x
2
1+
47
x
+
1
x
2
.
Now as x approaches in?nity, all the quotients with some power of x in the denominator
approach zero, leaving 2 in the numerator and 1 in the denominator, so the limit again is
2.
EXAMPLE 4.10.8 Compute lim
x?0
secx-1
sinx
.
Both the numerator and denominator approach zero, so applying L’Hˆ opital’s Rule:
lim
x?0
secx-1
sinx
= lim
x?0
secxtanx
cosx
=
1·0
1
= 0.
EXAMPLE 4.10.9 Compute lim
x?0
+
xlnx.
This doesn’t appear to be suitable for L’Hˆ opital’s Rule, but it also is not “obvious”.
As x approaches zero, lnx goes to-8, so the product looks like (something very small)·
(something very large and negative). But this could be anything: it depends on how small
and how large. Forexample, consider (x
2
)(1/x), (x)(1/x), and (x)(1/x
2
). Asx approaches
zero, each of these is (something very small)· (something very large), yet the limits are
respectively zero, 1, and8.
We can in fact turn this into a L’Hˆ opital’s Rule problem:
xlnx =
lnx
1/x
=
lnx
x
-1
.
Now as x approaches zero, both the numerator and denominator approach in?nity (one
-8 and one +8, but only the size is important). Using L’Hˆ opital’s Rule:
lim
x?0
+
lnx
x
-1
= lim
x?0
+
1/x
-x
-2
= lim
x?0
+
1
x
(-x
2
) = lim
x?0
+
-x = 0.
One way to interpret this is that since lim
x?0
+
xlnx = 0, the x approaches zero much faster
than the lnx approaches-8.
Page 5


4.10 Limits revisited We have de?ned and used the concept of limit, primarily in our development of the deriva-
tive. Recall that lim
x?a
f(x) = L is true if, in a precise sense, f(x) gets closer and closer to
L as x gets closer and closer to a. While some limits are easy to see, others take some
ingenuity; in particular, the limits that de?ne derivatives are always di?cult on their face,
since in
lim
?x?0
f(x+?x)-f(x)
?x
both the numerator and denominator approach zero. Typically this di?culty can be re-
solved when f isa“nice” function and we are tryingtocompute a derivative. Occasionally
such limits are interesting for other reasons, and the limit of a fraction in which both nu-
merator and denominator approach zero can be di?cult to analyze. Now that we have
the derivative available, there is another technique that can sometimes be helpful in such
circumstances.
Before we introduce the technique, we will also expand our concept of limit, in two
ways. When the limit of f(x) as x approaches a does not exist, it may be useful to note in
what way it does not exist. We have already talked about one such case: one-sided limits.
Another case is when “f goes to in?nity”. We also will occasionally want to know what
happens to f when x “goes to in?nity”.
EXAMPLE 4.10.1 What happens to 1/x as x goes to 0? From the right, 1/x gets
bigger and bigger, or goes to in?nity. From the left it goes to negative in?nity.
EXAMPLE 4.10.2 What happens to the function cos(1/x) as x goes to in?nity? It
seems clear that as x gets larger and larger, 1/x gets closer and closer to zero, so cos(1/x)
should be getting closer and closer to cos(0) = 1.
As with ordinary limits, these concepts can be made precise. Roughly, we want
lim
x?a
f(x) =8 to mean that we can make f(x) arbitrarily large by making x close enough
to a, and lim
x?8
f(x) = L should mean we can make f(x) as close as we want to L by
making x large enough. Compare this de?nition to the de?nition of limit in section 2.3,
de?nition 2.3.2.
DEFINITION 4.10.3 If f is a function, we say that lim
x?a
f(x) =8 if for every N > 0
there is a d > 0 such that whenever |x-a| < d, f(x) > N. We can extend this in the
obvious ways to de?ne lim
x?a
f(x)=-8, lim
x?a
-
f(x)=±8, and lim
x?a
+
f(x)=±8.
DEFINITION4.10.4 Limitatin?nity Iff isafunction,wesaythat lim
x?8
f(x) = L
if for every o > 0 there is an N > 0 so that whenever x > N, |f(x)-L| < o. We may
similarly de?ne lim
x?-8
f(x) = L, and using the idea of the previous de?nition, we may
de?ne lim
x?±8
f(x)=±8.
We include these de?nitions for completeness, but we will not explore them in detail.
Su?ce it to say that such limits behave in much the same way that ordinary limits do; in
particular there are some analogs of theorem 2.3.6.
Now consider this limit:
lim
x?p
x
2
-p
2
sinx
.
As x approaches p, both the numerator and denominator approach zero, so it is not
obvious what, if anything, the quotient approaches. We can often compute such limits by
application of the following theorem.
THEOREM 4.10.5 L’Hˆ opital’s Rule For “su?ciently nice” functions f(x) and
g(x), if lim
x?a
f(x) = 0 = lim
x?a
g(x) or both lim
x?a
f(x) =±8 and lim
x?a
g(x) =±8, and if
lim
x?a
f
'
(x)
g
'
(x)
exists, then lim
x?a
f(x)
g(x)
= lim
x?a
f
'
(x)
g
'
(x)
. This remains true if “x?a” is replaced by
“x?8” or “x?-8”.
This theorem is somewhat di?cult to prove, in part because it incorporates so many
di?erent possibilities, so we will not prove it here. We also will not need to worry about
the precise de?nition of “su?ciently nice”, as the functions we encounter will be suitable.
EXAMPLE 4.10.6 Compute lim
x?p
x
2
-p
2
sinx
in two ways.
First we use L’Hˆ opital’s Rule: Since the numerator and denominator both approach
zero,
lim
x?p
x
2
-p
2
sinx
= lim
x?p
2x
cosx
,
provided the latter exists. But in fact this is an easy limit, since the denominator now
approaches-1, so
lim
x?p
x
2
-p
2
sinx
=
2p
-1
=-2p.
We don’t really need L’Hˆ opital’s Rule to do this limit. Rewrite it as
lim
x?p
(x+p)
x-p
sinx
and note that
lim
x?p
x-p
sinx
= lim
x?p
x-p
-sin(x-p)
= lim
x?0
-
x
sinx
since x-p approaches zero as x approaches p. Now
lim
x?p
(x+p)
x-p
sinx
= lim
x?p
(x+p) lim
x?0
-
x
sinx
= 2p(-1) =-2p
as before.
EXAMPLE 4.10.7 Compute lim
x?8
2x
2
-3x+7
x
2
+47x+1
in two ways.
As x goes to in?nity both the numerator and denominator go to in?nity, so we may
apply L’Hˆ opital’s Rule:
lim
x?8
2x
2
-3x+7
x
2
+47x+1
= lim
x?8
4x-3
2x+47
.
In the second quotient, it is still the case that the numerator and denominator both go to
in?nity, so we are allowed to use L’Hˆ opital’s Rule again:
lim
x?8
4x-3
2x+47
= lim
x?8
4
2
= 2.
So the original limit is 2 as well.
Again, we don’t really need L’Hˆ opital’s Rule, and in fact a more elementary approach
is easier—we divide the numerator and denominator by x
2
:
lim
x?8
2x
2
-3x+7
x
2
+47x+1
= lim
x?8
2x
2
-3x+7
x
2
+47x+1
1
x
2
1
x
2
= lim
x?8
2-
3
x
+
7
x
2
1+
47
x
+
1
x
2
.
Now as x approaches in?nity, all the quotients with some power of x in the denominator
approach zero, leaving 2 in the numerator and 1 in the denominator, so the limit again is
2.
EXAMPLE 4.10.8 Compute lim
x?0
secx-1
sinx
.
Both the numerator and denominator approach zero, so applying L’Hˆ opital’s Rule:
lim
x?0
secx-1
sinx
= lim
x?0
secxtanx
cosx
=
1·0
1
= 0.
EXAMPLE 4.10.9 Compute lim
x?0
+
xlnx.
This doesn’t appear to be suitable for L’Hˆ opital’s Rule, but it also is not “obvious”.
As x approaches zero, lnx goes to-8, so the product looks like (something very small)·
(something very large and negative). But this could be anything: it depends on how small
and how large. Forexample, consider (x
2
)(1/x), (x)(1/x), and (x)(1/x
2
). Asx approaches
zero, each of these is (something very small)· (something very large), yet the limits are
respectively zero, 1, and8.
We can in fact turn this into a L’Hˆ opital’s Rule problem:
xlnx =
lnx
1/x
=
lnx
x
-1
.
Now as x approaches zero, both the numerator and denominator approach in?nity (one
-8 and one +8, but only the size is important). Using L’Hˆ opital’s Rule:
lim
x?0
+
lnx
x
-1
= lim
x?0
+
1/x
-x
-2
= lim
x?0
+
1
x
(-x
2
) = lim
x?0
+
-x = 0.
One way to interpret this is that since lim
x?0
+
xlnx = 0, the x approaches zero much faster
than the lnx approaches-8.
50. The function f(x) =
x
v
x
2
+1
has two horizontal asymptotes. Find them and give a rough
sketch of f with its horizontal asymptotes. ?
4.11 Hyperboli
 Fun
tions Thehyperbolicfunctions appearwithsomefrequency inapplications, andarequitesimilar
in many respects to the trigonometric functions. This is a bit surprising given our initial
de?nitions.
DEFINITION 4.11.1 The hyperbolic cosine is the function
coshx =
e
x
+e
-x
2
,
and the hyperbolic sine is the function
sinhx =
e
x
-e
-x
2
.
Notice that cosh is even (that is, cosh(-x) = cosh(x)) while sinh is odd (sinh(-x) =
-sinh(x)), and coshx+sinhx = e
x
. Also, for all x, coshx > 0, while sinhx = 0 if and
only if e
x
-e
-x
= 0, which is true precisely when x = 0.
LEMMA 4.11.2 The range of coshx is [1,8).
Proof. Let y = coshx. We solve for x:
y =
e
x
+e
-x
2
2y =e
x
+e
-x
2ye
x
=e
2x
+1
0 = e
2x
-2ye
x
+1
e
x
=
2y±
p
4y
2
-4
2
e
x
=y±
p
y
2
-1
From the last equation, we see y
2
= 1, and since y= 0, it follows that y= 1.
Now suppose y= 1, so y±
p
y
2
-1 > 0. Then x = ln(y±
p
y
2
-1) is a real number,
and y = coshx, so y is in the range of cosh(x).