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GATE 2026 Chemical Engineering, Q22 · Mathematics

The correct answer is A, −1.

Question

Which one of the following is the value of lim (x→0) (eˣ − x − 1) / (cos x − 1)?

  • A. −1
  • B. 0
  • C. 1
  • D. ∞\infty

Official answer: −1

Why

GATE 2026 official key: (A). This is a 0/0 form, so expand via Taylor series near x=0: eˣ−x−1 ≈x2/2\approx x²/2 and cos x−1 ≈−x2/2,\approx −x²/2, so the limit is (x2/2)/(−x2/2)=−1(x²/2)/(−x²/2) = −1 (equivalently, apply L'Hôpital's rule twice).

Explanation cross-checked for consistency. Final answer matches the official IIT answer key.

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