4.8 L’Hôpital’s Rule - Calculus Volume 1 | OpenStax (2024)

Learning Objectives

  • 4.8.1Recognize when to apply L’Hôpital’s rule.
  • 4.8.2Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case.
  • 4.8.3Describe the relative growth rates of functions.

In this section, we examine a powerful tool for evaluating limits. This tool, known as L’Hôpital’s rule, uses derivatives to calculate limits. With this rule, we will be able to evaluate many limits we have not yet been able to determine. Instead of relying on numerical evidence to conjecture that a limit exists, we will be able to show definitively that a limit exists and to determine its exact value.

Applying L’Hôpital’s Rule

L’Hôpital’s rule can be used to evaluate limits involving the quotient of two functions. Consider

limxaf(x)g(x).limxaf(x)g(x).

If limxaf(x)=L1andlimxag(x)=L20,limxaf(x)=L1andlimxag(x)=L20, then

limxaf(x)g(x)=L1L2.limxaf(x)g(x)=L1L2.

However, what happens if limxaf(x)=0limxaf(x)=0 and limxag(x)=0?limxag(x)=0? We call this one of the indeterminate forms, of type 00.00. This is considered an indeterminate form because we cannot determine the exact behavior of f(x)g(x)f(x)g(x) as xaxa without further analysis. We have seen examples of this earlier in the text. For example, consider

limx2x24x2andlimx0sinxx.limx2x24x2andlimx0sinxx.

For the first of these examples, we can evaluate the limit by factoring the numerator and writing

limx2x24x2=limx2(x+2)(x2)x2=limx2(x+2)=2+2=4.limx2x24x2=limx2(x+2)(x2)x2=limx2(x+2)=2+2=4.

For limx0sinxxlimx0sinxx we were able to show, using a geometric argument, that

limx0sinxx=1.limx0sinxx=1.

Here we use a different technique for evaluating limits such as these. Not only does this technique provide an easier way to evaluate these limits, but also, and more important, it provides us with a way to evaluate many other limits that we could not calculate previously.

The idea behind L’Hôpital’s rule can be explained using local linear approximations. Consider two differentiable functions ff and gg such that limxaf(x)=0=limxag(x)limxaf(x)=0=limxag(x) and such that g(a)0g(a)0 For xx near a,a, we can write

f(x)f(a)+f(a)(xa)f(x)f(a)+f(a)(xa)

and

g(x)g(a)+g(a)(xa).g(x)g(a)+g(a)(xa).

Therefore,

f(x)g(x)f(a)+f(a)(xa)g(a)+g(a)(xa).f(x)g(x)f(a)+f(a)(xa)g(a)+g(a)(xa).

4.8 L’Hôpital’s Rule - Calculus Volume 1 | OpenStax (1)

Figure 4.71 If limxaf(x)=limxag(x),limxaf(x)=limxag(x), then the ratio f(x)/g(x)f(x)/g(x) is approximately equal to the ratio of their linear approximations near a.a.

Since ff is differentiable at a,a, then ff is continuous at a,a, and therefore f(a)=limxaf(x)=0.f(a)=limxaf(x)=0. Similarly, g(a)=limxag(x)=0.g(a)=limxag(x)=0. If we also assume that ff and gg are continuous at x=a,x=a, then f(a)=limxaf(x)f(a)=limxaf(x) and g(a)=limxag(x).g(a)=limxag(x). Using these ideas, we conclude that

limxaf(x)g(x)=limxaf(x)(xa)g(x)(xa)=limxaf(x)g(x).limxaf(x)g(x)=limxaf(x)(xa)g(x)(xa)=limxaf(x)g(x).

Note that the assumption that ff and gg are continuous at aa and g(a)0g(a)0 can be loosened. We state L’Hôpital’s rule formally for the indeterminate form 00.00. Also note that the notation 0000 does not mean we are actually dividing zero by zero. Rather, we are using the notation 0000 to represent a quotient of limits, each of which is zero.

Theorem 4.12

L’Hôpital’s Rule (0/0 Case)

Suppose ff and gg are differentiable functions over an open interval containing a,a, except possibly at a.a. If limxaf(x)=0limxaf(x)=0 and limxag(x)=0,limxag(x)=0, then

limxaf(x)g(x)=limxaf(x)g(x),limxaf(x)g(x)=limxaf(x)g(x),

assuming the limit on the right exists or is or .. This result also holds if we are considering one-sided limits, or if a=and.a=and.

Proof

We provide a proof of this theorem in the special case when f,g,f,f,g,f, and gg are all continuous over an open interval containing a.a. In that case, since limxaf(x)=0=limxag(x)limxaf(x)=0=limxag(x) and ff and gg are continuous at a,a, it follows that f(a)=0=g(a).f(a)=0=g(a). Therefore,

limxaf(x)g(x)=limxaf(x)f(a)g(x)g(a)sincef(a)=0=g(a)=limxaf(x)f(a)xag(x)g(a)xaalgebra=limxaf(x)f(a)xalimxag(x)g(a)xalimit of a quotient=f(a)g(a)definition of the derivative=limxaf(x)limxag(x)continuity offandg=limxaf(x)g(x).limit of a quotientlimxaf(x)g(x)=limxaf(x)f(a)g(x)g(a)sincef(a)=0=g(a)=limxaf(x)f(a)xag(x)g(a)xaalgebra=limxaf(x)f(a)xalimxag(x)g(a)xalimit of a quotient=f(a)g(a)definition of the derivative=limxaf(x)limxag(x)continuity offandg=limxaf(x)g(x).limit of a quotient

Note that L’Hôpital’s rule states we can calculate the limit of a quotient fgfg by considering the limit of the quotient of the derivatives fg.fg. It is important to realize that we are not calculating the derivative of the quotient fg.fg.

Example 4.38

Applying L’Hôpital’s Rule (0/0 Case)

Evaluate each of the following limits by applying L’Hôpital’s rule.

  1. limx01cosxxlimx01cosxx
  2. limx1sin(πx)lnxlimx1sin(πx)lnx
  3. limxe1/x11/xlimxe1/x11/x
  4. limx0sinxxx2limx0sinxxx2

Solution

  1. Since the numerator 1cosx01cosx0 and the denominator x0,x0, we can apply L’Hôpital’s rule to evaluate this limit. We have

    limx01cosxx=limx0ddx(1cosx)ddx(x)=limx0sinx1=limx0(sinx)limx0(1)=01=0.limx01cosxx=limx0ddx(1cosx)ddx(x)=limx0sinx1=limx0(sinx)limx0(1)=01=0.

  2. As x1,x1, the numerator sin(πx)0sin(πx)0 and the denominator ln(x)0.ln(x)0. Therefore, we can apply L’Hôpital’s rule. We obtain

    limx1sin(πx)lnx=limx1πcos(πx)1/x=limx1(πx)cos(πx)=(π·1)(−1)=π.limx1sin(πx)lnx=limx1πcos(πx)1/x=limx1(πx)cos(πx)=(π·1)(−1)=π.

  3. As x,x, the numerator e1/x10e1/x10 and the denominator (1x)0.(1x)0. Therefore, we can apply L’Hôpital’s rule. We obtain

    limxe1/x11x=limxe1/x(−1x2)(−1x2)=limxe1/x=e0=1.limxe1/x11x=limxe1/x(−1x2)(−1x2)=limxe1/x=e0=1.

  4. As x0,x0, both the numerator and denominator approach zero. Therefore, we can apply L’Hôpital’s rule. We obtain

    limx0sinxxx2=limx0cosx12x.limx0sinxxx2=limx0cosx12x.


    Since the numerator and denominator of this new quotient both approach zero as x0,x0, we apply L’Hôpital’s rule again. In doing so, we see that

    limx0cosx12x=limx0sinx2=0.limx0cosx12x=limx0sinx2=0.


    Therefore, we conclude that

    limx0sinxxx2=0.limx0sinxxx2=0.

Checkpoint 4.37

Evaluate limx0xtanx.limx0xtanx.

We can also use L’Hôpital’s rule to evaluate limits of quotients f(x)g(x)f(x)g(x) in which f(x)±f(x)± and g(x)±.g(x)±. Limits of this form are classified as indeterminate forms of type /./. Again, note that we are not actually dividing by .. Since is not a real number, that is impossible; rather, /./. is used to represent a quotient of limits, each of which is or ..

Theorem 4.13

L’Hôpital’s Rule (/(/ Case)

Suppose ff and gg are differentiable functions over an open interval containing a,a, except possibly at a.a. Suppose limxaf(x)=limxaf(x)= (or )) and limxag(x)=limxag(x)= (or ).). Then,

limxaf(x)g(x)=limxaf(x)g(x),limxaf(x)g(x)=limxaf(x)g(x),

assuming the limit on the right exists or is or .. This result also holds if the limit is infinite, if a=a= or ,, or the limit is one-sided.

Example 4.39

Applying L’Hôpital’s Rule (/(/ Case)

Evaluate each of the following limits by applying L’Hôpital’s rule.

  1. limx3x+52x+1limx3x+52x+1
  2. limx0+lnxcotxlimx0+lnxcotx

Solution

  1. Since 3x+53x+5 and 2x+12x+1 are first-degree polynomials with positive leading coefficients, limx(3x+5)=limx(3x+5)= and limx(2x+1)=.limx(2x+1)=. Therefore, we apply L’Hôpital’s rule and obtain

    limx3x+52x+1/x=limx32=32.limx3x+52x+1/x=limx32=32.


    Note that this limit can also be calculated without invoking L’Hôpital’s rule. Earlier in the chapter we showed how to evaluate such a limit by dividing the numerator and denominator by the highest power of xx in the denominator. In doing so, we saw that

    limx3x+52x+1=limx3+5/x2+1/x=32.limx3x+52x+1=limx3+5/x2+1/x=32.


    L’Hôpital’s rule provides us with an alternative means of evaluating this type of limit.
  2. Here, limx0+lnx=limx0+lnx= and limx0+cotx=.limx0+cotx=. Therefore, we can apply L’Hôpital’s rule and obtain

    limx0+lnxcotx=limx0+1/xcsc2x=limx0+1xcsc2x.limx0+lnxcotx=limx0+1/xcsc2x=limx0+1xcsc2x.


    Now as x0+,x0+, csc2x.csc2x. Therefore, the first term in the denominator is approaching zero and the second term is getting really large. In such a case, anything can happen with the product. Therefore, we cannot make any conclusion yet. To evaluate the limit, we use the definition of cscxcscx to write

    limx0+1xcsc2x=limx0+sin2xx.limx0+1xcsc2x=limx0+sin2xx.


    Now limx0+sin2x=0limx0+sin2x=0 and limx0+x=0,limx0+x=0, so we apply L’Hôpital’s rule again. We find

    limx0+sin2xx=limx0+2sinxcosx−1=0−1=0.limx0+sin2xx=limx0+2sinxcosx−1=0−1=0.


    We conclude that

    limx0+lnxcotx=0.limx0+lnxcotx=0.

Checkpoint 4.38

Evaluate limxlnx5x.limxlnx5x.

As mentioned, L’Hôpital’s rule is an extremely useful tool for evaluating limits. It is important to remember, however, that to apply L’Hôpital’s rule to a quotient f(x)g(x),f(x)g(x), it is essential that the limit of f(x)g(x)f(x)g(x) be of the form 0000 or /./. Consider the following example.

Example 4.40

When L’Hôpital’s Rule Does Not Apply

Consider limx1x2+53x+4.limx1x2+53x+4. Show that the limit cannot be evaluated by applying L’Hôpital’s rule.

Solution

Because the limits of the numerator and denominator are not both zero and are not both infinite, we cannot apply L’Hôpital’s rule. If we try to do so, we get

d d x ( x 2 + 5 ) = 2 x d d x ( x 2 + 5 ) = 2 x

and

d d x ( 3 x + 4 ) = 3 . d d x ( 3 x + 4 ) = 3 .

At which point we would conclude erroneously that

lim x 1 x 2 + 5 3 x + 4 = lim x 1 2 x 3 = 2 3 . lim x 1 x 2 + 5 3 x + 4 = lim x 1 2 x 3 = 2 3 .

However, since limx1(x2+5)=6limx1(x2+5)=6 and limx1(3x+4)=7,limx1(3x+4)=7, we actually have

lim x 1 x 2 + 5 3 x + 4 = 6 7 . lim x 1 x 2 + 5 3 x + 4 = 6 7 .

We can conclude that

lim x 1 x 2 + 5 3 x + 4 lim x 1 d d x ( x 2 + 5 ) d d x ( 3 x + 4 ) . lim x 1 x 2 + 5 3 x + 4 lim x 1 d d x ( x 2 + 5 ) d d x ( 3 x + 4 ) .

Checkpoint 4.39

Explain why we cannot apply L’Hôpital’s rule to evaluate limx0+cosxx.limx0+cosxx. Evaluate limx0+cosxxlimx0+cosxx by other means.

Other Indeterminate Forms

L’Hôpital’s rule is very useful for evaluating limits involving the indeterminate forms 0000 and /./. However, we can also use L’Hôpital’s rule to help evaluate limits involving other indeterminate forms that arise when evaluating limits. The expressions 0·,0·, ,, 1,1, 0,0, and 0000 are all considered indeterminate forms. These expressions are not real numbers. Rather, they represent forms that arise when trying to evaluate certain limits. Next we realize why these are indeterminate forms and then understand how to use L’Hôpital’s rule in these cases. The key idea is that we must rewrite the indeterminate forms in such a way that we arrive at the indeterminate form 0000 or /./.

Indeterminate Form of Type 0·0·

Suppose we want to evaluate limxa(f(x)·g(x)),limxa(f(x)·g(x)), where f(x)0f(x)0 and g(x)g(x) (or )) as xa.xa. Since one term in the product is approaching zero but the other term is becoming arbitrarily large (in magnitude), anything can happen to the product. We use the notation 0·0· to denote the form that arises in this situation. The expression 0·0· is considered indeterminate because we cannot determine without further analysis the exact behavior of the product f(x)g(x)f(x)g(x) as xa.xa. For example, let nn be a positive integer and consider

f(x)=1(xn+1)andg(x)=3x2.f(x)=1(xn+1)andg(x)=3x2.

As x,x, f(x)0f(x)0 and g(x).g(x). However, the limit as xx of f(x)g(x)=3x2(xn+1)f(x)g(x)=3x2(xn+1) varies, depending on n.n. If n=2,n=2, then limxf(x)g(x)=3.limxf(x)g(x)=3. If n=1,n=1, then limxf(x)g(x)=.limxf(x)g(x)=. If n=3,n=3, then limxf(x)g(x)=0.limxf(x)g(x)=0. Here we consider another limit involving the indeterminate form 0·0· and show how to rewrite the function as a quotient to use L’Hôpital’s rule.

Example 4.41

Indeterminate Form of Type 0·0·

Evaluate limx0+xlnx.limx0+xlnx.

Solution

First, rewrite the function xlnxxlnx as a quotient to apply L’Hôpital’s rule. If we write

x ln x = ln x 1 / x , x ln x = ln x 1 / x ,

we see that lnxlnx as x0+x0+ and 1x1x as x0+.x0+. Therefore, we can apply L’Hôpital’s rule and obtain

lim x 0 + ln x 1 / x = lim x 0 + d d x ( ln x ) d d x ( 1 / x ) = lim x 0 + 1 / x -1 / x 2 = lim x 0 + ( x ) = 0 . lim x 0 + ln x 1 / x = lim x 0 + d d x ( ln x ) d d x ( 1 / x ) = lim x 0 + 1 / x -1 / x 2 = lim x 0 + ( x ) = 0 .

We conclude that

lim x 0 + x ln x = 0 . lim x 0 + x ln x = 0 .

Checkpoint 4.40

Evaluate limx0xcotx.limx0xcotx.

Indeterminate Form of Type

Another type of indeterminate form is .. Consider the following example. Let nn be a positive integer and let f(x)=3xnf(x)=3xn and g(x)=3x2+5.g(x)=3x2+5. As x,x, f(x)f(x) and g(x).g(x). We are interested in limx(f(x)g(x)).limx(f(x)g(x)). Depending on whether f(x)f(x) grows faster, g(x)g(x) grows faster, or they grow at the same rate, as we see next, anything can happen in this limit. Since f(x)f(x) and g(x),g(x), we write to denote the form of this limit. As with our other indeterminate forms, has no meaning on its own and we must do more analysis to determine the value of the limit. For example, suppose the exponent nn in the function f(x)=3xnf(x)=3xn is n=3,n=3, then

limx(f(x)g(x))=limx(3x33x25)=.limx(f(x)g(x))=limx(3x33x25)=.

On the other hand, if n=2,n=2, then

limx(f(x)g(x))=limx(3x23x25)=−5.limx(f(x)g(x))=limx(3x23x25)=−5.

However, if n=1,n=1, then

limx(f(x)g(x))=limx(3x3x25)=.limx(f(x)g(x))=limx(3x3x25)=.

Therefore, the limit cannot be determined by considering only .. Next we see how to rewrite an expression involving the indeterminate form as a fraction to apply L’Hôpital’s rule.

Example 4.42

Indeterminate Form of Type

Evaluate limx0+(1x21tanx).limx0+(1x21tanx).

Solution

By combining the fractions, we can write the function as a quotient. Since the least common denominator is x2tanx,x2tanx, we have

1 x 2 1 tan x = ( tan x ) x 2 x 2 tan x . 1 x 2 1 tan x = ( tan x ) x 2 x 2 tan x .

As x0+,x0+, the numerator tanxx20tanxx20 and the denominator x2tanx0.x2tanx0. Therefore, we can apply L’Hôpital’s rule. Taking the derivatives of the numerator and the denominator, we have

lim x 0 + ( tan x ) x 2 x 2 tan x = lim x 0 + ( sec 2 x ) 2 x x 2 sec 2 x + 2 x tan x . lim x 0 + ( tan x ) x 2 x 2 tan x = lim x 0 + ( sec 2 x ) 2 x x 2 sec 2 x + 2 x tan x .

As x0+,x0+, (sec2x)2x1(sec2x)2x1 and x2sec2x+2xtanx0.x2sec2x+2xtanx0. Since the denominator is positive as xx approaches zero from the right, we conclude that

lim x 0 + ( sec 2 x ) 2 x x 2 sec 2 x + 2 x tan x = . lim x 0 + ( sec 2 x ) 2 x x 2 sec 2 x + 2 x tan x = .

Therefore,

lim x 0 + ( 1 x 2 1 tan x ) = . lim x 0 + ( 1 x 2 1 tan x ) = .

Checkpoint 4.41

Evaluate limx0+(1x1sinx).limx0+(1x1sinx).

Another type of indeterminate form that arises when evaluating limits involves exponents. The expressions 00,00, 0,0, and 11 are all indeterminate forms. On their own, these expressions are meaningless because we cannot actually evaluate these expressions as we would evaluate an expression involving real numbers. Rather, these expressions represent forms that arise when finding limits. Now we examine how L’Hôpital’s rule can be used to evaluate limits involving these indeterminate forms.

Since L’Hôpital’s rule applies to quotients, we use the natural logarithm function and its properties to reduce a problem evaluating a limit involving exponents to a related problem involving a limit of a quotient. For example, suppose we want to evaluate limxaf(x)g(x)limxaf(x)g(x) and we arrive at the indeterminate form 0.0. (The indeterminate forms 0000 and 11 can be handled similarly.) We proceed as follows. Let

y=f(x)g(x).y=f(x)g(x).

Then,

lny=ln(f(x)g(x))=g(x)ln(f(x)).lny=ln(f(x)g(x))=g(x)ln(f(x)).

Therefore,

limxa[ln(y)]=limxa[g(x)ln(f(x))].limxa[ln(y)]=limxa[g(x)ln(f(x))].

Since limxaf(x)=,limxaf(x)=, we know that limxaln(f(x))=.limxaln(f(x))=. Therefore, limxag(x)ln(f(x))limxag(x)ln(f(x)) is of the indeterminate form 0·,0·, and we can use the techniques discussed earlier to rewrite the expression g(x)ln(f(x))g(x)ln(f(x)) in a form so that we can apply L’Hôpital’s rule. Suppose limxag(x)ln(f(x))=L,limxag(x)ln(f(x))=L, where LL may be or .. Then

limxa[ln(y)]=L.limxa[ln(y)]=L.

Since the natural logarithm function is continuous, we conclude that

ln(limxay)=L,ln(limxay)=L,

which gives us

limxay=limxaf(x)g(x)=eL.limxay=limxaf(x)g(x)=eL.

Example 4.43

Indeterminate Form of Type 00

Evaluate limxx1/x.limxx1/x.

Solution

Let y=x1/x.y=x1/x. Then,

ln ( x 1 / x ) = 1 x ln x = ln x x . ln ( x 1 / x ) = 1 x ln x = ln x x .

We need to evaluate limxlnxx.limxlnxx. Applying L’Hôpital’s rule, we obtain

lim x ln y = lim x ln x x = lim x 1 / x 1 = 0 . lim x ln y = lim x ln x x = lim x 1 / x 1 = 0 .

Therefore, limxlny=0.limxlny=0. Since the natural logarithm function is continuous, we conclude that

ln ( lim x y ) = 0 , ln ( lim x y ) = 0 ,

which leads to

lim x y = lim x ln x x = e 0 = 1 . lim x y = lim x ln x x = e 0 = 1 .

Hence,

lim x x 1 / x = 1 . lim x x 1 / x = 1 .

Checkpoint 4.42

Evaluate limxx1/ln(x).limxx1/ln(x).

Example 4.44

Indeterminate Form of Type 0000

Evaluate limx0+xsinx.limx0+xsinx.

Solution

Let

y = x sin x . y = x sin x .

Therefore,

ln y = ln ( x sin x ) = sin x ln x . ln y = ln ( x sin x ) = sin x ln x .

We now evaluate limx0+sinxlnx.limx0+sinxlnx. Since limx0+sinx=0limx0+sinx=0 and limx0+lnx=,limx0+lnx=, we have the indeterminate form 0·.0·. To apply L’Hôpital’s rule, we need to rewrite sinxlnxsinxlnx as a fraction. We could write

sin x ln x = sin x 1 / ln x sin x ln x = sin x 1 / ln x

or

sin x ln x = ln x 1 / sin x = ln x csc x . sin x ln x = ln x 1 / sin x = ln x csc x .

Let’s consider the first option. In this case, applying L’Hôpital’s rule, we would obtain

lim x 0 + sin x ln x = lim x 0 + sin x 1 / ln x = lim x 0 + cos x −1 / ( x ( ln x ) 2 ) = lim x 0 + ( x ( ln x ) 2 cos x ) . lim x 0 + sin x ln x = lim x 0 + sin x 1 / ln x = lim x 0 + cos x −1 / ( x ( ln x ) 2 ) = lim x 0 + ( x ( ln x ) 2 cos x ) .

Unfortunately, we not only have another expression involving the indeterminate form 0·,0·, but the new limit is even more complicated to evaluate than the one with which we started. Instead, we try the second option. By writing

sin x ln x = ln x 1 / sin x = ln x csc x , sin x ln x = ln x 1 / sin x = ln x csc x ,

and applying L’Hôpital’s rule, we obtain

lim x 0 + sin x ln x = lim x 0 + ln x csc x = lim x 0 + 1 / x csc x cot x = lim x 0 + −1 x csc x cot x . lim x 0 + sin x ln x = lim x 0 + ln x csc x = lim x 0 + 1 / x csc x cot x = lim x 0 + −1 x csc x cot x .

Using the fact that cscx=1sinxcscx=1sinx and cotx=cosxsinx,cotx=cosxsinx, we can rewrite the expression on the right-hand side as

lim x 0 + sin 2 x x cos x = lim x 0 + [ sin x x · ( tan x ) ] = ( lim x 0 + sin x x ) · ( lim x 0 + ( tan x ) ) = 1 · 0 = 0 . lim x 0 + sin 2 x x cos x = lim x 0 + [ sin x x · ( tan x ) ] = ( lim x 0 + sin x x ) · ( lim x 0 + ( tan x ) ) = 1 · 0 = 0 .

We conclude that limx0+lny=0.limx0+lny=0. Therefore, ln(limx0+y)=0ln(limx0+y)=0 and we have

lim x 0 + y = lim x 0 + x sin x = e 0 = 1 . lim x 0 + y = lim x 0 + x sin x = e 0 = 1 .

Hence,

lim x 0 + x sin x = 1 . lim x 0 + x sin x = 1 .

Checkpoint 4.43

Evaluate limx0+xx.limx0+xx.

Growth Rates of Functions

Suppose the functions ff and gg both approach infinity as x.x. Although the values of both functions become arbitrarily large as the values of xx become sufficiently large, sometimes one function is growing more quickly than the other. For example, f(x)=x2f(x)=x2 and g(x)=x3g(x)=x3 both approach infinity as x.x. However, as shown in the following table, the values of x3x3 are growing much faster than the values of x2.x2.

xx10101001001000100010,00010,000
f(x)=x2f(x)=x210010010,00010,0001,000,0001,000,000100,000,000100,000,000
g(x)=x3g(x)=x3100010001,000,0001,000,0001,000,000,0001,000,000,0001,000,000,000,0001,000,000,000,000

Table 4.7 Comparing the Growth Rates of x 2 x 2 and x 3 x 3

In fact,

limxx3x2=limxx=.or, equivalently,limxx2x3=limx1x=0.limxx3x2=limxx=.or, equivalently,limxx2x3=limx1x=0.

As a result, we say x3x3 is growing more rapidly than x2x2 as x.x. On the other hand, for f(x)=x2f(x)=x2 and g(x)=3x2+4x+1,g(x)=3x2+4x+1, although the values of g(x)g(x) are always greater than the values of f(x)f(x) for x>0,x>0, each value of g(x)g(x) is roughly three times the corresponding value of f(x)f(x) as x,x, as shown in the following table. In fact,

limxx23x2+4x+1=13.limxx23x2+4x+1=13.

xx10101001001000100010,00010,000
f(x)=x2f(x)=x210010010,00010,0001,000,0001,000,000100,000,000100,000,000
g(x)=3x2+4x+1g(x)=3x2+4x+134134130,40130,4013,004,0013,004,001300,040,001300,040,001

Table 4.8 Comparing the Growth Rates of x 2 x 2 and 3 x 2 + 4 x + 1 3 x 2 + 4 x + 1

In this case, we say that x2x2 and 3x2+4x+13x2+4x+1 are growing at the same rate as x.x.

More generally, suppose ff and gg are two functions that approach infinity as x.x. We say gg grows more rapidly than ff as xx if

limxg(x)f(x)=;or, equivalently,limxf(x)g(x)=0.limxg(x)f(x)=;or, equivalently,limxf(x)g(x)=0.

On the other hand, if there exists a constant M0M0 such that

limxf(x)g(x)=M,limxf(x)g(x)=M,

we say ff and gg grow at the same rate as x.x.

Next we see how to use L’Hôpital’s rule to compare the growth rates of power, exponential, and logarithmic functions.

Example 4.45

Comparing the Growth Rates of ln(x),ln(x), x2,x2, and exex

For each of the following pairs of functions, use L’Hôpital’s rule to evaluate limx(f(x)g(x)).limx(f(x)g(x)).

  1. f(x)=x2andg(x)=exf(x)=x2andg(x)=ex
  2. f(x)=ln(x)andg(x)=x2f(x)=ln(x)andg(x)=x2

Solution

  1. Since limxx2=limxx2= and limxex=,limxex=, we can use L’Hôpital’s rule to evaluate limx[x2ex].limx[x2ex]. We obtain

    limxx2ex=limx2xex.limxx2ex=limx2xex.


    Since limx2x=limx2x= and limxex=,limxex=, we can apply L’Hôpital’s rule again. Since

    limx2xex=limx2ex=0,limx2xex=limx2ex=0,


    we conclude that

    limxx2ex=0.limxx2ex=0.


    Therefore, exex grows more rapidly than x2x2 as xx (See Figure 4.73 and Table 4.9).
    4.8 L’Hôpital’s Rule - Calculus Volume 1 | OpenStax (3)

    Figure 4.73 An exponential function grows at a faster rate than a power function.

    xx55101015152020
    x2x22525100100225225400400
    exex14814822,02622,0263,269,0173,269,017485,165,195485,165,195

    Table 4.9 Growth rates of a power function and an exponential function.

  2. Since limxlnx=limxlnx= and limxx2=,limxx2=, we can use L’Hôpital’s rule to evaluate limxlnxx2.limxlnxx2. We obtain

    limxlnxx2=limx1/x2x=limx12x2=0.limxlnxx2=limx1/x2x=limx12x2=0.


    Thus, x2x2 grows more rapidly than lnxlnx as xx (see Figure 4.74 and Table 4.10).
    4.8 L’Hôpital’s Rule - Calculus Volume 1 | OpenStax (4)

    Figure 4.74 A power function grows at a faster rate than a logarithmic function.

    xx10101001001000100010,00010,000
    ln(x)ln(x)2.3032.3034.6054.6056.9086.9089.2109.210
    x2x210010010,00010,0001,000,0001,000,000100,000,000100,000,000

    Table 4.10 Growth rates of a power function and a logarithmic function

Checkpoint 4.44

Compare the growth rates of x100x100 and 2x.2x.

Using the same ideas as in Example 4.45a. it is not difficult to show that exex grows more rapidly than xpxp for any p>0.p>0. In Figure 4.75 and Table 4.11, we compare exex with x3x3 and x4x4 as x.x.

4.8 L’Hôpital’s Rule - Calculus Volume 1 | OpenStax (5)

Figure 4.75 The exponential function exex grows faster than xpxp for any p>0.p>0. (a) A comparison of exex with x3.x3. (b) A comparison of exex with x4.x4.

xx55101015152020
x3x3125125100010003375337580008000
x4x462562510,00010,00050,62550,625160,000160,000
exex14814822,02622,0263,269,0173,269,017485,165,195485,165,195

Table 4.11 An exponential function grows at a faster rate than any power function

Similarly, it is not difficult to show that xpxp grows more rapidly than lnxlnx for any p>0.p>0. In Figure 4.76 and Table 4.12, we compare lnxlnx with x3x3 and x.x.

4.8 L’Hôpital’s Rule - Calculus Volume 1 | OpenStax (6)

Figure 4.76 The function y=ln(x)y=ln(x) grows more slowly than xpxp for any p>0p>0 as x.x.

xx10101001001000100010,00010,000
ln(x)ln(x)2.3032.3034.6054.6056.9086.9089.2109.210
x3x32.1542.1544.6424.642101021.54421.544
xx3.1623.162101031.62331.623100100

Table 4.12 A logarithmic function grows at a slower rate than any root function

Section 4.8 Exercises

For the following exercises, evaluate the limit.

356.

Evaluate the limit limxexx.limxexx.

357.

Evaluate the limit limxexxk.limxexxk.

358.

Evaluate the limit limxlnxxk.limxlnxxk.

359.

Evaluate the limit limxaxax2a2,a0limxaxax2a2,a0.

360.

Evaluate the limit limxaxax3a3,a0limxaxax3a3,a0.

361.

Evaluate the limit limxaxaxnan,a0limxaxaxnan,a0.

For the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.

362.

lim x 0 + x 2 ln x lim x 0 + x 2 ln x

363.

lim x x 1 / x lim x x 1 / x

364.

lim x 0 x 2 / x lim x 0 x 2 / x

365.

lim x 0 x 2 1 / x lim x 0 x 2 1 / x

366.

lim x e x x lim x e x x

For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.

367.

lim x 3 x 2 9 x 3 lim x 3 x 2 9 x 3

368.

lim x 3 x 2 9 x + 3 lim x 3 x 2 9 x + 3

369.

lim x 0 ( 1 + x ) −2 1 x lim x 0 ( 1 + x ) −2 1 x

370.

lim x π / 2 cos x π 2 x lim x π / 2 cos x π 2 x

371.

lim x π x π sin x lim x π x π sin x

372.

lim x 1 x 1 sin x lim x 1 x 1 sin x

373.

lim x 0 ( 1 + x ) n 1 x lim x 0 ( 1 + x ) n 1 x

374.

lim x 0 ( 1 + x ) n 1 n x x 2 lim x 0 ( 1 + x ) n 1 n x x 2

375.

lim x 0 sin x tan x x 3 lim x 0 sin x tan x x 3

376.

lim x 0 1 + x 1 x x lim x 0 1 + x 1 x x

377.

lim x 0 e x x 1 x 2 lim x 0 e x x 1 x 2

378.

lim x 0 + tan x x lim x 0 + tan x x

379.

lim x 1 x 1 ln x lim x 1 x 1 ln x

380.

lim x 0 ( x + 1 ) 1 / x lim x 0 ( x + 1 ) 1 / x

381.

lim x 1 x x 3 x 1 lim x 1 x x 3 x 1

382.

lim x 0 + x 2 x lim x 0 + x 2 x

383.

lim x x sin ( 1 x ) lim x x sin ( 1 x )

384.

lim x 0 sin x x x 2 lim x 0 sin x x x 2

385.

lim x 0 + x ln ( x 4 ) lim x 0 + x ln ( x 4 )

386.

lim x ( x e x ) lim x ( x e x )

387.

lim x x 2 e x lim x x 2 e x

388.

lim x 0 3 x 2 x x lim x 0 3 x 2 x x

389.

lim x 0 1 + 1 / x 1 1 / x lim x 0 1 + 1 / x 1 1 / x

390.

lim x π / 4 ( 1 tan x ) cot x lim x π / 4 ( 1 tan x ) cot x

391.

lim x x e 1 / x lim x x e 1 / x

392.

lim x 0 + x 1 / cos x lim x 0 + x 1 / cos x

393.

lim x 0 + x 1 / x lim x 0 + x 1 / x

394.

lim x 0 - ( 1 1 x ) x lim x 0 - ( 1 1 x ) x

395.

lim x ( 1 1 x ) x lim x ( 1 1 x ) x

For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.

396.

[T] limx0ex1xlimx0ex1x

397.

[T] limx0xsin(1x)limx0xsin(1x)

398.

[T] limx1x11cos(πx)limx1x11cos(πx)

399.

[T] limx1e(x1)1x1limx1e(x1)1x1

400.

[T] limx1(x1)2lnxlimx1(x1)2lnx

401.

[T] limxπ1+cosxsinxlimxπ1+cosxsinx

402.

[T] limx0(cscx1x)limx0(cscx1x)

403.

[T] limx0+tan(xx)limx0+tan(xx)

404.

[T] limx0+lnxsinxlimx0+lnxsinx

405.

[T] limx0exexxlimx0exexx

4.8 L’Hôpital’s Rule - Calculus Volume 1 | OpenStax (2024)

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