---
title: "Limit of (sin x)^(tan x) as x→π"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:limit-sinx-power-tanx-at-pi"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/limit-sinx-power-tanx-at-pi
site: BCT Engineering
---

# Limit of (sin x)^(tan x) as x→π

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2071 Shrawan, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/limit-sinx-power-tanx-at-pi
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Evaluate lim_{x→π} (sin x)^(tan x).

## 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.

## Answer

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:

```text
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.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. (sin x)^(tan x) as x tends to pi is of the form:
   A. 1^infinity
   B. 0/0
   C. 0^0
   D. infinity^0
   Answer: C. Both sin and tan tend to 0. (Key point: Form)

2. The limit is taken as x tends to pi from the left because:
   A. for convenience
   B. tan x is undefined
   C. cos x is zero
   D. sin x is positive only for x < pi near pi
   Answer: D. Needed for the power to be real. (Key point: Side)

3. ln L = lim ln(sin x)/cot x is of the form:
   A. 0/0
   B. 0 * 0
   C. infinity/infinity
   D. 1^infinity
   Answer: C. ln(sin x) tends to -infinity and cot to infinity in magnitude. (Key point: Rewrite)

4. After L'Hospital, ln L = -lim cos x sin x, which is:
   A. 0, so L = 1
   B. infinity
   C. 1, so L = e
   D. -1
   Answer: A. sin pi = 0. (Key point: Result)

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