4.7 Using L'Hopitals Rule for Determining Limits in Indeterminate Forms

Use L’Hospital’s Rule only when the limit of a quotient gives an indeterminate form 0/0 or ∞/∞ after direct substitution. Before you apply it, check these CED conditions: the numerator and denominator are differentiable near the point (except possibly at the point), and the derivative of the denominator isn’t zero in a neighborhood (so the new quotient is defined). Then you can replace lim f(x)/g(x) with lim f′(x)/g′(x). You may need to apply it more than once if the result is still 0/0 or ∞/∞. Always try algebraic simplification (factor, expand, divide out common factors, or use dominant-term/series reasoning at infinity) before L’Hospital—sometimes that’s enough and faster. Note AP exams only assess 0/0 and ∞/∞ cases (other indeterminate forms like ∞−∞ aren’t tested on AB/BC per the CED). For review and examples, see the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM). For extra practice, check the unit overview (https://library.fiveable.me/ap-calculus/unit-4) and the AP practice pool (https://library.fiveable.me/practice/ap-calculus).

L’Hospital’s Rule (informal formula): If lim_{x→a} f(x) = 0 and lim_{x→a} g(x) = 0 (0/0), or both → ±∞ (∞/∞), and f and g are differentiable on an open interval around a (except maybe at a) with g′(x) ≠ 0 nearby, then lim_{x→a} f(x)/g(x) = lim_{x→a} f′(x)/g′(x), provided the right-hand limit exists (finite or ±∞). You can apply the rule repeatedly if f′/g′ still gives 0/0 or ∞/∞. It works for limits as x→a or x→±∞. Important cautions (from the CED): only use it for the indeterminate forms 0/0 or ∞/∞, verify differentiability and that g′ ≠ 0 near the point, and check the derivative-limit exists. For AP review and examples, see the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM) and Unit 4 overview (https://library.fiveable.me/ap-calculus/unit-4). For extra practice, try problems at (https://library.fiveable.me/practice/ap-calculus).

0/0 is called an indeterminate form—it doesn’t mean the limit is 0 or undefined in a simple way. It just tells you that both numerator and denominator are going to 0, so you can’t tell the limit’s value by direct substitution. The limit might be 0, a finite nonzero number, ∞, or fail to exist. How you resolve it: - Try algebraic simplification (factor/cancel) first. Example: lim_{x→1} (x^2−1)/(x−1) → cancel (x−1) → limit = 2. - If simplification doesn’t help and the functions meet the differentiability conditions, use L’Hospital’s Rule: replace lim f/g with lim f′/g′ (can repeat if still indeterminate). Remember the CED requirement: L’Hospital applies for 0/0 or ∞/∞ when numerator and denominator are differentiable near the point (except possibly at the point itself). For extra practice and AP-aligned examples, check the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM) and plenty of practice problems (https://library.fiveable.me/practice/ap-calculus).

Yes—infinity over infinity is an indeterminate form you can use L’Hôpital’s Rule on. The CED explicitly lists 0/0 and ∞/∞ as indeterminate forms that may be evaluated with L’Hôpital’s Rule (LIM-4.A.1–2). Conditions you must check first: - As x → a (or ±∞) the numerator and denominator both tend to ∞ (or both to 0). - Both functions are differentiable on a neighborhood (except possibly at a) and the derivative of the denominator is not 0 near that point. If those hold, then lim f(x)/g(x) = lim f′(x)/g′(x), provided the right-hand limit exists (or is ±∞). You can apply the rule repeatedly if the new quotient is still 0/0 or ∞/∞. Quick example: lim_{x→∞} x / e^x → both ∞, differentiate once to get lim_{x→∞} 1 / e^x = 0. For AP: the exam asks you to recognize and use L’Hôpital for 0/0 and ∞/∞ (Topic 4.7). For more examples and a short study guide, see the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM). For lots of practice problems, check the practice page (https://library.fiveable.me/practice/ap-calculus).

Step-by-step for a 0/0 limit using L’Hôpital’s Rule: 1. Confirm indeterminate: plug in the limit point. If you get 0/0 (or ∞/∞), L’Hôpital can apply; otherwise don’t use it. (CED: LIM-4.A.1) 2. Check conditions: numerator and denominator must be differentiable near the point (except maybe at the point itself). (CED: differentiability requirement) 3. Differentiate top and bottom separately: replace f(x)/g(x) by f′(x)/g′(x). Don’t use the quotient rule on the whole fraction—differentiate numerator and denominator individually. (CED: LIM-4.A.2) 4. Evaluate the new limit. If it gives a finite number or ±∞, that’s your answer. 5. If you still get 0/0 (or ∞/∞), repeat L’Hôpital—you may need higher-order derivatives. Stop when you get a determinate form or an answer that diverges. 6. Always consider algebraic simplification first (factoring, conjugates)—sometimes simplification avoids calculus. This is what AP expects: verify indeterminate form, ensure differentiability, apply derivatives, repeat as needed. For extra practice and examples see the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM) and the Unit 4 overview (https://library.fiveable.me/ap-calculus/unit-4). For many practice problems, try (https://library.fiveable.me/practice/ap-calculus).

Plugging in a number is the first thing you should try: if direct substitution gives a finite value (like 5 or −2 or ∞ in an obvious non-indeterminate way), that’s the limit. L’Hôpital’s Rule only applies when substitution yields an indeterminate quotient 0/0 or ∞/∞ (CED LIM-4.A.1). In those cases you can replace lim f(x)/g(x) by lim f′(x)/g′(x) provided f and g are differentiable near the point (CED LIM-4.A.2). Key practical points: - Step 1: substitute. If you get a number, you’re done. - Step 2: if you get 0/0 or ∞/∞, use L’Hôpital’s Rule (maybe more than once) or try algebraic simplification first—sometimes factoring or conjugates is simpler and faster. - Remember the differentiability requirement and that L’Hôpital doesn’t apply to other indeterminate forms (CED exclusion). For more examples and AP-style practice, check the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM) and Unit 4 resources (https://library.fiveable.me/ap-calculus/unit-4). For lots of practice problems, see (https://library.fiveable.me/practice/ap-calculus).

Think of a limit as asking “what number does this fraction get close to as x approaches a value?” - 5/0 (numerator nonzero, denominator → 0) is undefined because the fraction blows up—it doesn’t approach a finite number. Depending on the sign of the denominator it goes to +∞ or −∞ (or doesn’t settle), so there’s no finite limit. This is not an indeterminate form used with L’Hôpital. - 0/0 is indeterminate because both numerator and denominator are vanishing, and different functions that give 0/0 can have different limiting values. Example: lim (x→0) x/x = 1, but lim (x→0) x^2/x = 0, and lim (x→0) x/(|x|) does not exist. So knowing you have 0/0 gives no information about the limit’s value—it’s “indeterminate.” That’s why the CED/LIM-4.A says L’Hôpital’s Rule applies when a limit gives 0/0 or ∞/∞: those forms let you differentiate numerator and denominator to try to resolve the ambiguity (see the Topic 4.7 study guide for examples: https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM). For more practice, use the AP Calc practice bank (https://library.fiveable.me/practice/ap-calculus).

Take the derivative of the numerator and the derivative of the denominator separately—you do NOT differentiate the whole fraction as one object. L’Hôpital’s Rule says: if as x → a the quotient f(x)/g(x) gives the indeterminate form 0/0 or ∞/∞ (CED LIM-4.A.1–4.A.2), then provided f and g are differentiable near a and g′(x) ≠ 0, you may replace the original limit with lim x→a f′(x)/g′(x). If that new limit is still indeterminate, you can apply the rule again (repeated application/higher-order derivatives). Always check the conditions first (forms and differentiability) and simplify/algebra first if possible. For AP-specific review and examples see the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM) and try extra practice at (https://library.fiveable.me/practice/ap-calculus).

You can apply L’Hospital’s Rule as many times as needed—but only while the conditions hold and the limit stays in an indeterminate 0/0 or ∞/∞ form. Each time you apply it you must check: - Condition: numerator and denominator both approach 0 or both approach ±∞ at the same limit point (CED LIM-4.A.1). - Differentiability: both functions are differentiable near the point (CED: derivative requirement). - New form: after taking derivatives, evaluate the new limit. If it’s still 0/0 or ∞/∞, you may apply L’Hospital again (CED keywords: repeated application, higher-order derivatives). - Stop when the limit evaluates to a finite number or ±∞, or when the form is no longer 0/0 or ∞/∞. If repeated differentiation doesn’t resolve the indeterminacy, try algebraic simplification, series expansion, or substitution instead. On the AP exam you won’t be asked other indeterminate forms (like ∞−∞) for scoring, but teachers may include them. For a focused review, see the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM) and more unit resources (https://library.fiveable.me/ap-calculus/unit-4). For lots of practice, try the AP problems at (https://library.fiveable.me/practice/ap-calculus).

You must only apply L’Hôpital’s Rule when the limit gives the indeterminate forms 0/0 or ∞/∞ (CED LIM-4.A.1–4.A.2). If the expression is NOT one of those: - If numerator → nonzero finite and denominator → 0, the limit is infinite (±∞) or does not exist—L’Hôpital doesn’t apply. - If numerator → finite a and denominator → finite b with b ≠ 0, the limit is a/b—no need for L’Hôpital. - If numerator → ∞ and denominator → finite nonzero, the limit is ±∞—again L’Hôpital is inappropriate. - If you get another indeterminate form (∞ − ∞, 0·∞, 0^0, etc.), you must first algebraically rewrite it into 0/0 or ∞/∞ (or use another test); note the CED excludes many of those from exam assessment. Also check hypotheses: the functions must be differentiable near the point and denominator’s derivative shouldn’t be zero everywhere. Only repeat L’Hôpital when the quotient of derivatives still yields 0/0 or ∞/∞. For more AP-aligned review, see the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

You check that direct substitution gives 0/0, so it’s an indeterminate form (LIM-4.A.1). L’Hospital’s Rule (LIM-4.A.2) applies because numerator and denominator are differentiable near 0 and the limit of their ratio is 0/0. Differentiate top and bottom: d/dx[sin x] = cos x, d/dx[x] = 1. So by L’Hospital, lim_{x→0} (sin x)/x = lim_{x→0} (cos x)/1 = cos 0 = 1. That’s all you need for the AP: identify the 0/0 form, verify differentiability, apply L’Hospital once, and evaluate the resulting limit. For more on L’Hospital problems and AP-style practice, see the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM) and extra practice (https://library.fiveable.me/practice/ap-calculus).

Short answer: an indeterminate form is a specific algebraic signal (like 0/0 or ∞/∞) that tells you substitution didn't give the limit—it doesn’t tell you the limit’s value. “Doesn’t exist” (DNE) is the actual outcome when a limit has no finite value or left/right limits disagree. Why it matters: - Indeterminate (0/0 or ∞/∞): you must do more work (algebraic simplification, factoring, series, or L’Hôpital’s Rule—and remember the derivative conditions) because the limit could be any finite number, 0, or ±∞. Example: lim x→0 (sin x)/x -> substitution gives 0/0 (indeterminate) but the limit = 1. AP exams only test 0/0 and ∞/∞ for L’Hôpital (CED LIM-4.A). - DNE: either left and right limits differ, or it truly blows up (e.g., lim x→0 1/x → ±∞ or two-sided limits unequal). No amount of L’Hôpital will turn a clearly divergent one-sided mismatch into a finite limit. If you want step-by-step examples and practice (important for the AP), check the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM) and practice problems (https://library.fiveable.me/practice/ap-calculus).

Short answer: yes—you should always check that the limit gives an indeterminate form 0/0 or ∞/∞ before applying L’Hôpital’s Rule. Why: L’Hôpital’s Rule (CED LIM-4.A.2) only guarantees that lim f/g = lim f′/g′ when the original ratio tends to 0/0 or ∞/∞ and f and g are differentiable near the point (except possibly at the point itself). If the limit is a finite nonzero number, or simply ∞, or another non-indeterminate form, L’Hôpital doesn’t apply and using it can give wrong answers. Also check differentiability of numerator and denominator near the limit and consider one-sided limits (if the problem asks). If you get 0/0 or ∞/∞ after the first application you may repeat L’Hôpital’s Rule (repeated application/higher-order derivatives). If algebraic simplification or substitution removes the indeterminate form first, do that—it’s often quicker on the AP. For a quick refresher, see the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM). For more practice, check the Unit 4 overview (https://library.fiveable.me/ap-calculus/unit-4) and the AP practice problems (https://library.fiveable.me/practice/ap-calculus).

Yes—L’Hôpital’s Rule works when the limit is as x → ∞ (or x → −∞), not just when x → a finite number. The CED’s LIM-4.A covers both indeterminate forms 0/0 and ∞/∞—those can occur as x → a or as x → ±∞. Requirements: the original limit must produce 0/0 or ∞/∞, the numerator and denominator are differentiable on an open interval (except possibly at the point/infinite end), and the limit of f '(x)/g '(x) exists (or you can repeat the rule). If that derivative-limit exists (finite or ±∞), it equals the original limit—useful for finding horizontal asymptotes. On the AP exam, show the indeterminate form, state L’Hôpital’s conditions, differentiate numerator and denominator, and justify any repeated application. For a quick refresher and examples from Unit 4.7, see the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM). For extra practice, try problems at Fiveable’s practice page (https://library.fiveable.me/practice/ap-calculus).

Your calculator shows an error because it’s doing a direct numeric operation: 0/0 is undefined arithmeticly, so the device can’t give a number. But in limit language 0/0 is an indeterminate form—it tells you more work is needed, not the final answer. L’Hôpital’s Rule (CED LIM-4.A) lets you replace the original limit of f(x)/g(x) (when both → 0 or both → ∞) by the limit of f′(x)/g′(x), provided the derivatives exist near the point and that latter limit exists (or you can apply the rule again). That analytic process finds the behavior of the ratio as x approaches the point; a calculator doing plain substitution can’t do that reasoning. For more on when and how to apply L’Hôpital (including repeated application and differentiability requirements), see the Topic 4.7 study guide (https://library.fiveable.me/ap-calculus/unit-4/using-lhopitals-rule-for-determining-limits-indeterminate-forms/study-guide/8TvE04DRvJ3Z5PG2indM). For extra practice, check the Unit 4 review (https://library.fiveable.me/ap-calculus/unit-4) and the practice bank (https://library.fiveable.me/practice/ap-calculus).