---
title: "Limit of (tan x / x)^(1/x²)"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:limit-tanx-over-x-power-1-over-x2"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/limit-tanx-over-x-power-1-over-x2
site: BCT Engineering
---

# Limit of (tan x / x)^(1/x²)

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2074 Asoj; 2070 Asar (asked 2 times)
- Video: [L'Hospital Rule - (tanx/x)^1/x^2](https://www.youtube.com/watch?v=SUogqR2lIMY) — Confused Student
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/limit-tanx-over-x-power-1-over-x2
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Evaluate lim_{x→0} (tan x / x)^(1/x²).

## Short answer

Let L = lim_{x→0} (tan x/x)^{1/x²}. Then ln L = lim ln(tan x/x)/x² = 1/3 (by series or L'Hôpital). Hence L = e^{1/3}.

## Answer

Let L = lim_{x→0} (tan x / x)^(1/x²). This is the form 1^∞.

ln L = lim_{x→0} ln(tan x / x) / x² = lim [ln tan x − ln x] / x²  (form 0/0)

Apply L'Hospital's rule:

```text
ln L = lim [ sec²x/tan x − 1/x ] / (2x)
     = lim [ 2/sin 2x − 1/x ] / (2x)
     = lim (2x − sin 2x) / (2x² sin 2x)
```
Use series sin 2x = 2x − (2x)³/6 + ... = 2x − 4x³/3 + ...

Numerator: 2x − sin 2x = 4x³/3 + ...
Denominator: 2x² · 2x = 4x³ + ...

ln L = (4/3)/4 = 1/3.

(Direct series check: tan x / x = 1 + x²/3 + ..., so ln(tan x / x) = x²/3 + ..., and dividing by x² gives 1/3.)

**Answer: L = e^(1/3).**

## Check yourself

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

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

2. tan x / x = 1 + x^2/3 + ..., so ln(tan x / x) is about:
   A. x^2/3
   B. x/3
   C. x^2
   D. 3x^2
   Answer: A. ln(1 + u) ~ u. (Key point: Series)

3. Dividing x^2/3 by x^2 gives ln L =
   A. 3
   B. 0
   C. 1
   D. 1/3
   Answer: D. ln L = 1/3. (Key point: Value of ln L)

4. L equals:
   A. e
   B. e^3
   C. 1
   D. e^(1/3)
   Answer: D. L = e^(1/3). (Key point: Final result)

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