ITLR - Indeterminate Forms and L'Hopital's Rule Lesson

Indeterminate Forms and L'Hopital's Rule

Identifying an Indeterminate Form

An indeterminate form is a product, quotient, difference, or power of functions that are undefined when the argument of the function has a certain value, because one or both of the functions are zero or infinite. The indeterminate forms include $0^0,0/0,1^{\infty},\infty-\infty,\infty/ \infty ,0.\infty , \text{&}\infty ^{0}$.  These forms arise frequently in the context of finding limits. Given $\lim_{x \to \infty }\frac{3x-1}{x+1}$, the result is $\infty /\infty$, which is resolved by dividing the numerator and denominator by x and evaluated as shown.

$\lim_{x \to \infty }\frac{3x-1}{x+1}= \lim_{x \to \infty }\frac{\frac{3x}{x}-\frac{1}{x}}{\frac{x}{x}+\frac{1}{x}}= \lim_{x \to \infty }\frac{\lim_{x \to \infty }3-\lim_{x \to \infty }\frac{1}{x}}{\lim_{x \to \infty }1+\lim_{x \to \infty }\frac{1}{x}}=\frac{3-\frac{1}{x}}{1+\frac{1}{x}}=\frac{3-0}{1-0}=3$

However, algebraic manipulation of some indeterminate forms is not always successful. For example, direct substitution in the $\lim_{x \to 0}\frac{e^{3x}-1}{x}$ produces the indeterminate form 0/0, and rewriting the limit as $\lim_{x \to \infty }\left(\frac{e^{3x}}{x}-\frac{1}{x}\right)$yields $\infty-\infty$, another indeterminate form. Even the usual algebraic procedure of dividing by the highest power of x gives exactly the same result as the original limit.  Another tactic is to use technology to estimate the limit graphically and numerically.  Based on the graph and table, it appears as though the limiting value is 3, although graphs and tables are not definitive proof.

 

Using L'Hopital's Rule

L'Hopital's Rule is a technique for evaluating indeterminate forms of the type 0/0 or  .  The rule says that if the limit of a quotient of differentiable functions on an open interval (a, b) containing c, except possibly at c itself, produces the indeterminate form 0/0 or , then the limit of the quotient of their derivatives, provided the derivative of the denominator does not equal 0, is given by

$\lim_{x \to c}\frac{f\left(x\right)}{gx}=\lim_{x \to c}\frac{f'\left(x\right)}{g'\left(x\right)}$

A proof of L'Hopital's Rule is accessible from the More Resources sidebar section. The presentation below develops the rationale for L'Hopital's Rule and applies it to a familiar limit problem, $\lim_{x \to 0}\frac{\sin x}{x}$.

View the presentation introducing L'Hopital's Rule with five accompanying examples of its use.

There are times when L'Hopital's Rule can be applied but other methods may work just as well.  The next presentation compares the use of L'Hopital's Rule with factoring to evaluate limits.

It is not uncommon for L'Hopital's Rule to be applied multiple times when evaluating a limit.  The video below illustrates the application of L'Hopital's Rule three times.

The next two presentations provide additional examples of how and when to apply L'Hopital's Rule.

Explore L'Hopital's Rule by CLICKING HERE Links to an external site..

Aside from the power and usefulness of L'Hopital's Rule, danger exists when it is incorrectly applied to the limit of any rational function, whether or not direct substitution produces 0/0 or $\pm \infty /\pm \infty$, or when f '(x)/g'(x) is mistakenly assumed to be the derivative of a quotient.

Indeterminate Forms and L'Hopital's Rule Practice

Indeterminate Forms and L'Hopital's Rule: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of indeterminate forms and L'Hopital's Rule.

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