Indeterminate forms occur when direct substitution into a limit gives an expression that does not reveal the limit's value. Common examples include 0/0, ∞/∞, 0·∞, ∞ - ∞, 0^0, 1^∞, and ∞^0. They matter because the same form can lead to many different answers, including 0, a finite number, ∞, or no limit.
Calculus gives systematic tools for turning these unclear expressions into forms that can be evaluated.
Key Facts
- 0/0 and ∞/∞ are classic cases for L'Hopital's Rule when its conditions are met: lim f(x)/g(x) = lim f'(x)/g'(x).
- A 0·∞ form can often be rewritten as a quotient: f(x)g(x) = f(x)/(1/g(x)) or g(x)/(1/f(x)).
- An ∞ - ∞ form often needs algebra first, such as combining fractions, rationalizing, or using a common denominator.
- Power forms 0^0, 1^∞, and ∞^0 are often handled by setting y = f(x)^g(x) and taking ln y = g(x)ln(f(x)).
- Direct substitution giving 0/0 or ∞/∞ does not mean the limit is undefined, only that more analysis is needed.
- Useful techniques include factoring, canceling, rationalizing, common denominators, trig identities, series approximations, and L'Hopital's Rule.
Vocabulary
- Indeterminate form
- An expression from direct substitution in a limit that does not determine a unique limit value by itself.
- L'Hopital's Rule
- A theorem that allows certain 0/0 or ∞/∞ limits of quotients to be found by differentiating the numerator and denominator separately.
- Direct substitution
- The method of evaluating a limit by plugging the approaching input value into the function when the function behaves continuously there.
- Rationalizing
- An algebraic technique that multiplies by a conjugate to remove radicals or reveal canceling factors.
- Dominant term
- The term in an expression that has the greatest effect on the value as the variable approaches a given point or infinity.
Common Mistakes to Avoid
- Treating 0/0 as equal to 1 is wrong because numerator and denominator may approach 0 at different rates, giving many possible limit values.
- Using L'Hopital's Rule on any difficult limit is wrong because the rule applies only to quotients with 0/0 or ∞/∞ forms and requires differentiability conditions.
- Canceling terms across addition or subtraction is wrong because cancellation is valid for common factors, not separate terms in sums or differences.
- Stopping after seeing ∞ - ∞ is wrong because this form is indeterminate and may become a clear limit after combining fractions or rationalizing.
Practice Questions
- 1 Evaluate lim x→2 (x^2 - 4)/(x - 2).
- 2 Evaluate lim x→0 sin(5x)/(3x).
- 3 Explain why lim x→∞ (sqrt(x^2 + x) - x) has the indeterminate form ∞ - ∞ and describe a method that can resolve it.