---
title: "Limit of (2−x)^(tan(πx/2))"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:limit-2-minus-x-power-tan"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/limit-2-minus-x-power-tan
site: BCT Engineering
---

# Limit of (2−x)^(tan(πx/2))

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2074 Chaitra, 4 marks
- Video: [Find the limit of (2-a/x)^tan(pix/2a) in hindi](https://www.youtube.com/watch?v=XqFm9EFu7kE) — AdnanKhanAligYT
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/limit-2-minus-x-power-tan
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Evaluate lim_{x→1} (2−x)^(tan(πx/2)).

## Short answer

Form 1^∞. Take ln: ln L = lim tan(pi x/2) ln(2-x) = lim ln(2-x)/cot(pi x/2). Apply L'Hospital to get ln L = 2/pi, so L = e^(2/pi).

## Answer

Let L = lim_{x→1} (2 − x)^(tan(πx/2)). At x = 1: 2 − x → 1 and tan(πx/2) → ∞, so this is the form 1^∞.

Take logarithms:

ln L = lim tan(πx/2) · ln(2 − x) = lim ln(2 − x) / cot(πx/2)  (form 0/0)

Apply L'Hospital's rule:

```text
ln L = lim [−1/(2−x)] / [−(π/2) cosec²(πx/2)]
     = lim (2/π) · sin²(πx/2) / (2 − x)
     = (2/π) · 1/1 = 2/π
```
**Answer: L = e^(2/π).**

## Check yourself

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

1. (2-x)^(tan(pi x/2)) as x tends to 1 is of the form:
   A. infinity^0
   B. 0/0 directly
   C. 0^0
   D. 1^infinity
   Answer: D. Base tends to 1; tan tends to infinity. (Key point: Form)

2. ln L is rewritten as:
   A. ln(2-x) * tan which is 0 * 0
   B. ln(x)
   C. ln(2-x)/cot(pi x/2) of type 0/0
   D. tan/ln
   Answer: C. tan = 1/cot. (Key point: Rewrite)

3. The derivative ratio simplifies to:
   A. (2/pi) sin^2(pi x/2)/(2-x)
   B. (pi/2) sin^2/(2-x)
   C. 1/(2-x)
   D. (2/pi) cos^2
   Answer: A. Numerator -1/(2-x), denominator -(pi/2) cosec^2. (Key point: L'Hospital)

4. L =
   A. 1
   B. e^(pi/2)
   C. e
   D. e^(2/pi)
   Answer: D. ln L = 2/pi. (Key point: Result)

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