Indeterminate Forms & L’Hôpital’s Rule – Key Ideas

Flashcards cover definitions, theorems, conversion tricks, and example calculations related to indeterminate forms and L’Hôpital’s Rule.

What is an indeterminate form in the context of limits?

An algebraic form (e.g., 0/0, ∞/∞) in which the limit cannot be determined directly and may take infinitely many values depending on the functions involved.

List the seven classic indeterminate forms.

0/0, ∞/∞, 0·∞, ∞ − ∞, 0^0, ∞^0, 1^∞.

State L’Hôpital’s Rule for the 0/0 form.

If limₓ→a f(x)=0 and limₓ→a g(x)=0 with f and g differentiable near a, and g'(x)≠0 near a, then limₓ→a f(x)/g(x) = limₓ→a f'(x)/g'(x) provided the latter limit exists or is ±∞.

State L’Hôpital’s Rule for the ∞/∞ form.

If limₓ→a |f(x)|=∞ and limₓ→a |g(x)|=∞ with f and g differentiable near a, then limₓ→a f(x)/g(x)=limₓ→a f'(x)/g'(x) provided the derivative limit exists or is ±∞.

What should you do if L’Hôpital’s Rule yields another indeterminate form?

Apply L’Hôpital’s Rule repeatedly until the limit is determinate or use algebraic manipulation to simplify.

When should factoring or algebraic simplification be used instead of L’Hôpital’s Rule?

When the expression can be easily rewritten to eliminate the indeterminate form without differentiation (e.g., simple factoring, rationalization).

Convert the form 0·∞ into a quotient suitable for L’Hôpital’s Rule.

Rewrite as f(x)·g(x)=f(x)/(1/g(x)) or g(x)/(1/f(x)), producing 0/0 or ∞/∞.

Convert the form ∞ − ∞ into a quotient suitable for L’Hôpital’s Rule.

Combine terms over a common denominator or factor out the dominant term to create 0/0 or ∞/∞.

Evaluate limₓ→∞ x²/eˣ using L’Hôpital’s Rule.

Applying the rule twice gives limₓ→∞ 2/eˣ = 0, so the limit is 0.

Evaluate limₓ→π/6 [ln (cos 3x)] / [ln (tan 3x)].

Repeated application of L’Hôpital’s Rule gives −1.

What derivative is often used when applying L’Hôpital’s Rule to logarithmic expressions?

d/dx [ln u(x)] = u'(x)/u(x).

Why must g'(x) ≠ 0 near the point of interest when using L’Hôpital’s Rule?

To ensure the quotient f'(x)/g'(x) is well-defined and the rule is applicable.