Chapter 3: Applications of Differentiation
Section 3.9: Indeterminate Forms and L'Hôpital's Rule
Essentials
The seven collections of symbols on the left in Table 3.9.1 are called indeterminate forms, insofar as they arise when calculating the limits of certain expressions. The stratagems on the right in the table extend the set of rules for computing limits without having to invoke Definition 1.2.1.
Indeterminate Form
Limit Stratagem
00, ∞∞
Use L'Hôpital's rule
0⋅∞
Rewrite as fraction and use L'Hôpital's rule
∞−∞
Convert to quotient via common denominator, rationalization, or factoring; then use L'Hôpital's rule
00, ∞0, 1∞
Take the limit of the log and exponentiate the result.
Equivalently, write uxvx as ev lnu.
Table 3.9.1 Indeterminate forms and appropriate approaches to their limits
Recall from Table 1.3.1 that the limit of a quotient is the quotient of the limits when the limits exist and the limit of the denominator is not zero. If, in computing the limit of the fraction fx/gx one of the forms 0/0 or ∞/∞ results, no formal technique in Chapter 1 applies.
Recall from Table 1.3.1 that the limit of a product is the product of the limits, provided both limits exist. If, in computing the limit of the product fx⋅gx the form 0⋅∞ results, no formal technique in Chapter 1 applies.
Recall from Table 1.3.1 that the limit of a difference is the difference of the limits, provided both limits exist. If, in computing the limit of the difference fx−gx the form ∞−∞ results, no formal technique in Chapter 1 applies.
The three indeterminate forms at the end of Table 3.9.1 arise when taking the limit of expressions of the form uxvx. They do not arise if one of the functions u or v is strictly constant. For example, Maple evaluates 00 to 1, and limx→∞1x = 1.
Theorem 3.9.1 - L'Hôpital's Rule
a∈I, an open interval
f and g differentiable on I
g′x≠0 on I, except possibly at x=a
limx→afx=0 and limx→agx=0 or limx→afx=±∞ and limx→agx=±∞
limx→afxgx exists or is ±∞
⇒
limx→afxgx=limx→af ′xg′x
Condition 4 of Theorem 3.9.1 states that L'Hôpital's rule deals with the two indeterminate forms 0/0 and ∞/∞. If the rule is applied to a form other than one of these, an incorrect limit will be obtained.
If limx→afx−gx results in the indeterminate form ∞−∞, the algebra in Table 3.9.2 provides a formal way of converting the difference to an expression amenable to L'Hôpital's rule.
f−g≡1g−1f1f g
Table 3.9.2 Converting the form ∞−∞ to 0/0
Since f→∞ and g→∞, the numerator of the fraction on the right in Table 3.9.2 tends to 0−0=0. Similarly, the denominator on the right tends to 0 also. Hence, Table 3.9.2 converts the form ∞−∞ to 0/0 for which L'Hôpital's rule is applicable. However, the algebra involved is cumbersome, and any ad hoc manipulations that can replace the formalism of Table 3.9.2 should be considered.
Note on the pronunciation of "L'Hôpital"
There is variation in the spelling of this French name, some authors using the upper-case "L", some using the lower-case. Some texts even use the alternate spelling L'Hospital, causing students to pronounce the "s".
Consultation with a French mathematician yielded the following conclusions. Always use the upper-case "L". Never spell the name with an "s". Never pronounce the name with an "s". A typical North American pronunciation would be lōp-ē-tăhl'. The internet provides sites at which the French pronunciation can be heard.
Examples
Example 3.9.1
Evaluate limx→0sinxx, then detail an applicable strategy taken from Table 3.9.1.
Example 3.9.2
Evaluate limx→0sin2xx, then detail an applicable strategy taken from Table 3.9.1.
Example 3.9.3
Evaluate limx→1x2−1x−1, then detail an applicable strategy taken from Table 3.9.1.
Example 3.9.4
Evaluate limx→0x+1−1x, then detail an applicable strategy taken from Table 3.9.1.
Example 3.9.5
Evaluate limx→∞exx2, then detail an applicable strategy taken from Table 3.9.1.
Example 3.9.6
Evaluate limx→∞2 x2−13 x2+5 x, then detail an applicable strategy taken from Table 3.9.1.
Example 3.9.7
Evaluate limx→0+x lnx, then detail an applicable strategy taken from Table 3.9.1.
Example 3.9.8
Evaluate limx→0cscx−cotx, then detail an applicable strategy taken from Table 3.9.1.
Example 3.9.9
Evaluate limx→0+xx, then detail an applicable strategy taken from Table 3.9.1.
Example 3.9.10
Evaluate limx→0+−lnxx, then detail an applicable strategy taken from Table 3.9.1.
Example 3.9.11
Evaluate limx→∞x1+xx, then detail an applicable strategy taken from Table 3.9.1.
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