L’Hôpital’s Rule: Learn It 1

  • Spot indeterminate forms like in calculations, and use L’Hôpital’s rule to find precise values
  • Explain how quickly different functions increase or decrease compared to each other

L’Hôpital’s Rule

L’Hôpital’s Rule is a powerful technique for evaluating limits of functions. It leverages derivatives to determine limits that are otherwise challenging to compute directly, providing a clear way to resolve indeterminate forms.

Applying L’Hôpital’s Rule

L’Hôpital’s Rule comes into play when you’re dealing with limits that result in indeterminate forms.

Consider the quotient of two functions, represented as:

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

If limxaf(x)=L1limxaf(x)=L1 and limxag(x)=L20limxag(x)=L20, then

limxaf(x)g(x)=L1L2limxaf(x)g(x)=L1L2

However, what happens if limxaf(x)=0 and limxag(x)=0?

We call this one of the indeterminate forms, of type 00. This is considered an indeterminate form because we cannot determine the exact behavior of f(x)g(x) as xa without further analysis. We have seen examples of this earlier in the text.

  • limx2x24x2 can be solved by factoring the numerator and simplifying.
    limx2x24x2=limx2(x+2)(x2)x2=limx2(x+2)=2+2=4
  • limx0sinxx has been proven geometrically to equal 1. Using L’Hôpital’s Rule, differentiating both numerator and denominator yields:
    limx0sinxx=1

L’Hôpital’s Rule not only simplifies the calculation of certain limits but also provides insights into evaluating complex limits that would be difficult to handle by other methods. This rule is particularly useful in analyzing the behavior of functions as they approach critical points.

L’Hôpital’s Rule (0/0 Case)

The idea behind L’Hôpital’s rule can be explained using local linear approximations.

Consider two differentiable functions f and g such that limxaf(x)=0=limxag(x) and such that g(a)0 For x near a, we can write

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

and

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

Therefore,

f(x)g(x)f(a)+f(a)(xa)g(a)+g(a)(xa)
Two functions y = f(x) and y = g(x) are drawn such that they cross at a point above x = a. The linear approximations of these two functions y = f(a) + f’(a)(x – a) and y = g(a) + g’(a)(x – a) are also drawn.
Figure 1. If limxaf(x)=limxag(x), then the ratio f(x)/g(x) is approximately equal to the ratio of their linear approximations near a.

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

Using these ideas, we conclude that:

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

Note that the assumption that f and g are continuous at a and g(a)0 can be loosened.

The notation 00 does not mean we are actually dividing zero by zero. Rather, we are using the notation 00 to represent a quotient of limits, each of which is zero.

We state L’Hôpital’s rule formally for the indeterminate form 00

L’Hôpital’s rule (0/0 case)

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

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= or .

Proof


We provide a proof of this theorem in the special case when f,g,f, and g are all continuous over an open interval containing a. In that case, since limxaf(x)=0=limxag(x) and f and g are continuous at a, it follows that 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)xamultiplying numerator and denominator by1xa=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 fg by considering the limit of the quotient of the derivatives fg. It is important to realize that we are not calculating the derivative of the quotient fg.

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

  1. limx01cosxx
  2. limx1sin(πx)lnx
  3. limxe1x11x
  4. limx0sinxxx2