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Limit of (eˣ−1−x)/x² as x→0: Solved

Find the limit of (eˣ − 1 − x)/x² as x approaches 0. Direct substitution gives (1 − 1 − 0)/0 = 0/0, the classic indeterminate form. The expression is not broken; it is undecided. Beginners either declare it unsolvable or split the fraction into pieces that do not help.

The tool for 0/0 is l’Hôpital’s rule: differentiate the top and the bottom separately, then try the limit again. Here one round is not enough (the form is still 0/0), which is exactly where students stop. Two rounds finish it, and the answer is the exact fraction 1/2.

Worked solution

  1. Check the form at x = 0: numerator e⁰ − 1 − 0 = 0, denominator 0² = 0. The form is 0/0, so l’Hôpital’s rule applies.
  2. Differentiate the numerator and the denominator separately. This is not the quotient rule. d/dx[eˣ − 1 − x] = eˣ − 1 and d/dx[x²] = 2x. The new limit is (eˣ − 1)/(2x) as x → 0.
  3. Check the form again before doing anything else: (1 − 1)/0 = 0/0. Still indeterminate, so l’Hôpital applies a second time.
  4. Differentiate again: d/dx[eˣ − 1] = eˣ and d/dx[2x] = 2. The new limit is eˣ/2 as x → 0.
  5. Now substitute: e⁰/2 = 1/2. The limit is 1/2, exactly. (It equals 0.5, but 1/2 is the exact form markers expect.)

Why this method works

l’Hôpital’s rule replaces a stuck 0/0 ratio with the ratio of the derivatives, which for smooth functions like eˣ is a simpler comparison of growth rates. Each application peels off one layer of the indeterminacy: the first round removes the constant terms, the second removes the linear leftovers, and what remains is directly evaluable. It works because differentiating preserves the limiting behavior of 0/0 forms while stripping away the parts that caused the deadlock.

Common mistakes

  • Applying l’Hôpital without checking the form. The rule is legal only on 0/0 or ∞/∞. Check before every round, not just the first; a form can change between rounds.
  • Stopping after one round. (eˣ − 1)/(2x) is still 0/0. An answer of “0/0”, or a number guessed after one round, is wrong.
  • Using the quotient rule instead of differentiating top and bottom separately. l’Hôpital’s rule is not (f/g)′. This confusion is the single most common error on this topic.
  • Algebra slips in the derivatives. d/dx[eˣ − 1 − x] = eˣ − 1: the −1 is constant so it vanishes, and d/dx[−x] = −1.

Check (never skip)

Two independent checks confirm 1/2. Series check: eˣ = 1 + x + x²/2 + x³/6 + …, so eˣ − 1 − x = x²/2 + x³/6 + …, and dividing by x² gives 1/2 + x/6 + … → 1/2 as x → 0. ✓ Numeric check: at x = 0.001, e^0.001 ≈ 1.0010005002, so (1.0010005002 − 1 − 0.001)/0.000001 ≈ 0.5002, visibly closing in on 1/2. ✓

Seeing it numerically

Before touching l’Hôpital, a small table builds intuition for 1/2:

  • x = 0.1: e^0.1 ≈ 1.1051709, so (1.1051709 − 1 − 0.1)/0.01 = 0.5171
  • x = 0.01: e^0.01 ≈ 1.0100502, so (1.0100502 − 1 − 0.01)/0.0001 = 0.5017
  • x = 0.001: e^0.001 ≈ 1.0010005, so (1.0010005 − 1 − 0.001)/0.000001 = 0.5002

The values 0.517, 0.502, 0.500 march toward 1/2 from above. A table never proves a limit, but it tells you what to expect, and it would immediately expose an answer like 0 or 1 as wrong. Notice the approach is from above: every value slightly exceeds 1/2, because the next series term x/6 is positive for small positive x. That detail comes free from the series check, and it is the kind of observation that turns a computed answer into an understood one. Use the table to predict, l’Hôpital to prove.

What if the form is not 0/0?

l’Hôpital’s rule also covers ∞/∞, but this problem never goes there. If your first check gives something like 3/0, stop: that is not indeterminate, the limit is infinite (or does not exist), and l’Hôpital would give nonsense. The habit to build is check first, differentiate second, and that habit is worth more than any single answer.

FAQ

Q: When is l’Hôpital’s rule allowed? A: Only for 0/0 or ∞/∞ forms, with the numerator and denominator differentiable near (not necessarily at) the point. Verify the form before every application, because one round can change it.

Q: Why isn’t the answer 0? A: 0/0 does not mean 0; it means “not enough information yet.” Both parts vanish, but their ratio settles at 1/2, exactly as the series expansion shows.

Q: Can I use the Taylor series instead of l’Hôpital? A: Yes, and it doubles as the check above. l’Hôpital is usually faster under exam pressure; the series is more transparent about why the answer is 1/2.

Solve it with steps on our limit calculator.