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

# Limit of (2−x/a)^(tan(πx/2a))

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2070 Chaitra
- Video: [Evaluate Lt (2-x/a)^tan(π/2a) as x tends to a](https://www.youtube.com/watch?v=H_NUpG4s-0g) — Doctor of Mathematics
- 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-over-a-power-tan
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Evaluate lim_{x→a} (2 − x/a)^(tan(πx/2a)).

## Short answer

This is also form 1^∞. Take ln, apply L'Hospital: ln L = lim ln(2 - x/a)/cot(pi x/(2a)) = 2/pi. Thus L = e^(2/pi).

## Answer

Let L = lim_{x→a} (2 − x/a)^(tan(πx/2a)). At x = a the base → 1 and the exponent → ∞: form 1^∞.

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

Apply L'Hospital's rule. Derivative of the numerator: (−1/a)/(2 − x/a). Derivative of the denominator: −(π/2a) cosec²(πx/2a).

```text
ln L = lim [ (−1/a)/(2 − x/a) ] / [ −(π/2a) cosec²(πx/2a) ]
     = lim (2/π) sin²(πx/2a) / (2 − x/a)
     = (2/π)(1)/(1) = 2/π
```
**Answer: L = e^(2/π).** (Putting x = at reduces this to the previous problem, lim_{t→1} (2 − t)^(tan(πt/2)).)

## Check yourself

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

1. At x = a the base 2 - x/a and exponent give:
   A. infinity^0
   B. 0^0
   C. 0/0
   D. 1^infinity
   Answer: D. Base 1, exponent infinity. (Key point: Form)

2. d/dx of ln(2 - x/a) is:
   A. (-1/a)/(2 - x/a)
   B. -a/(2-x/a)
   C. 1/(2-x/a)
   D. (1/a)(2 - x/a)
   Answer: A. Chain rule. (Key point: Derivative of numerator)

3. At x = a, sin^2(pi x/2a) =
   A. 1
   B. pi
   C. 0
   D. 1/2
   Answer: A. sin(pi/2) = 1. (Key point: Sin factor)

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

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