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Concept Version 8
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Indeterminate Forms and L'Hôpital's Rule

Indeterminate forms like 00\frac{0}{0}​0​​0​​ have no definite value; however, when a limit is indeterminate, l'Hôpital's rule can often be used to evaluate it.

Learning Objective

  • Use L'Hopital's Rule to evaluate limits involving indeterminate forms


Key Points

    • Indeterminate forms include 000^00​0​​, 00\frac{0}{0}​0​​0​​, 1∞1^\infty1​∞​​, ∞−∞\infty - \infty∞−∞, ∞∞\frac{\infty}{\infty}​∞​​∞​​, 0×∞0 \times \infty0×∞, and ∞0\infty^0∞​0​​
    • Indeterminate forms often arise when you are asked to take the limit of a function. For example: limx→0xx\lim_{x\to 0}\frac{x}{x}lim​x→0​​​x​​x​​ is indeterminate, giving 00\frac{0}{0}​0​​0​​.
    • L'Hôpital's rule: For fff and ggg which are differentiable, if limx→cf(x)=limx→cg(x)=0\lim_{x\to c}f(x)=\lim_{x \to c}g(x) = 0lim​x→c​​f(x)=lim​x→c​​g(x)=0 or ±∞\pm \infty±∞ and limx→cf′(x)g′(x)\lim_{x\to c}\frac{f'(x)}{g'(x)}lim​x→c​​​g​′​​(x)​​f​′​​(x)​​ exists, and g′(x)≠0g'(x) \neq 0g​′​​(x)≠0 for all xxx in the interval containing ccc, then limx→cf(x)g(x)=limx→cf′(x)g′(x)\lim_{x \to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}lim​x→c​​​g(x)​​f(x)​​=lim​x→c​​​g​′​​(x)​​f​′​​(x)​​.

Terms

  • limit

    a value to which a sequence or function converges

  • differentiable

    a function that has a defined derivative (slope) at each point

  • indeterminate

    not accurately determined or determinable


Full Text

Occasionally in mathematics, one runs across an equation with an indeterminate form as seen in . In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is known as an indeterminate form. The indeterminate forms include 000^00​0​​, 00\frac{0}{0}​0​​0​​, 1∞1^\infty1​∞​​, ∞−∞\infty - \infty∞−∞, ∞∞\frac{\infty}{\infty}​∞​​∞​​, 0×∞0 \times \infty0×∞, and ∞0\infty^0∞​0​​.

The most common example of an indeterminate form is 00\frac{0}{0}​0​​0​​. As xxx approaches 000, the ratios xx3\frac{x}{x^3}​x​3​​​​x​​, xx\frac{x}{x}​x​​x​​, and x2x\frac{x^2}{x}​x​​x​2​​​​ go to ∞\infty∞, 111, and 000, respectively. In each case, however, if the limits of the numerator and denominator are evaluated and plugged into the division operation, the resulting expression is 00\frac{0}{0}​0​​0​​. So, roughly speaking, 00\frac{0}{0}​0​​0​​ can be 000, or ∞\infty∞, or it can be 111—in fact, it is possible to construct similar examples converging to any particular value. That is why the expression 00\frac{0}{0}​0​​0​​ is indeterminate.

More formally, the fact that the functions fff and ggg both approach 000 as xxx approaches some limit point ccc is not enough information to evaluate the limit limx→cf(x)g(x)\lim_{x\to c}\frac{f(x)}{g(x)}lim​x→c​​​g(x)​​f(x)​​. That limit could converge to any number, or diverge to infinity, or might not exist, depending on what the functions fff and ggg are. For example, limx→0xx\lim_{x\to 0}\frac{x}{x}lim​x→0​​​x​​x​​ is indeterminate.

L'Hôpital's Rule

In calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit.

In its simplest form, l'Hôpital's rule states that for functions fff and ggg which are differentiable, if 

limx→cf(x)=limx→cg(x)=0 or ±∞\displaystyle{\lim_{x\to c}f(x)=\lim_{x \to c}g(x) = 0 \text{ or } \pm \infty}​x→c​lim​​f(x)=​x→c​lim​​g(x)=0 or ±∞ 

and limx→cf′(x)g′(x)\lim_{x\to c}\frac{f'(x)}{g'(x)}lim​x→c​​​g​′​​(x)​​f​′​​(x)​​ exists, and g′(x)≠0g'(x) \neq 0g​′​​(x)≠0 for all xxx in the interval containing ccc, then: 

limx→cf(x)g(x)=limx→cf′(x)g′(x)\displaystyle{\lim_{x \to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}}​x→c​lim​​​g(x)​​f(x)​​=​x→c​lim​​​g​′​​(x)​​f​′​​(x)​​

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