- 2071 Shrawan · 5 marks
Limit of (sin x)^(tan x) as x→π
Short answer
Form 0^0 (limit from the left). Let L = lim, then ln L = lim tan x ln(sin x) = lim ln(sin x)/cot x. L'Hospital gives ln L = -lim cos x sin x = 0, so L = 1.
Evaluate lim_{x→π} (sin x)^(tan x).
Let L = lim_{x→π} (sin x)^(tan x). As x → π, sin x → 0 and tan x → 0: form 0⁰. (sin x is positive only for x < π, so the limit is taken as x → π⁻.)
ln L = lim tan x · ln(sin x) = lim ln(sin x)/cot x (form ∞/∞)
Apply L'Hospital's rule:
ln L = lim (cot x)/(−cosec²x)
= −lim cos x · sin x (since cot x / cosec²x = cos x sin x)
= −(−1)(0) = 0
Answer: L = e⁰ = 1.
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Limit of (tan x / x)^(1/x) · Mathematics I, chapter 1 · asked 3×
- Limit of (tan x / x)^(1/x²) · Mathematics I, chapter 1 · asked 2×
- Limit of (1/x²)^(tan x) · Mathematics I, chapter 1
- Limit of (2−x)^(tan(πx/2)) · Mathematics I, chapter 1
- Limit of (2−x/a)^(tan(πx/2a)) · Mathematics I, chapter 1
- Limit of (sin x / x)^(1/x) · Mathematics I, chapter 1 · asked 2×
- Triple product for r = a cos t i + a sin t j + at tanα k · Mathematics, chapter 5
- Residue theorem: ∮tan z dz on |z|=2 · Applied Mathematics, chapter 1
- Euler's theorem and tan⁻¹ function · Mathematics, chapter 1
- Evaluate a line integral F=sinyi+x(1+cosy)j · Mathematics III, chapter 2 · asked 6×