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

# Limit of (tan x / x)^(1/x)

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2072 Kartik; 2068 Shrawan; 2067 Asoj (asked 3 times)
- Video: [Evaluate lim(x→0) (tanx/x)^(1/x) || limit x tend to 0 (tanx/x)^1/x indeterminate form](https://www.youtube.com/watch?v=gB9VG9CVCU0) — XCKUENACMJG
- 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-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Evaluate lim_{x→0} (tan x / x)^(1/x).

## Short answer

Take natural log: ln L = lim_{x→0} [ln(tan x) − ln x]/x. Using L’Hôpital’s rule or series expansion, this limit is 0, so L = e^0 = 1.

## Answer

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

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

Apply L'Hospital's rule:

```text
ln L = lim [ sec²x / tan x − 1/x ]
     = lim [ 1/(sin x cos x) − 1/x ]
     = lim [ 2/sin 2x − 1/x ]
```
Using sin 2x = 2x − (2x)³/6 + ..., we get 2/sin 2x = (1/x)(1 + 2x²/3 + ...) = 1/x + 2x/3 + ...

So ln L = lim (1/x + 2x/3 − 1/x) = 0.

(Check by series: tan x / x = 1 + x²/3 + ..., so ln(tan x / x) ≈ x²/3 and divided by x gives x/3 → 0.)

**Answer: L = e⁰ = 1.**

## Check yourself

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

1. What form does (tan x / x)^(1/x) take as x tends to 0?
   A. 1^infinity
   B. 0/0
   C. infinity^0
   D. 0^0
   Answer: A. tan x / x tends to 1 while 1/x tends to infinity. (Key point: Indeterminate form)

2. What is done first to evaluate the limit?
   A. Apply L'Hospital directly to the power
   B. Square both sides
   C. Take ln, so ln L = lim [ln(tan x) - ln x]/x
   D. Replace tan x by x
   Answer: C. Taking logs converts the exponential 1^infinity form into a 0/0 quotient. (Key point: Logarithm trick)

3. Using tan x / x = 1 + x^2/3 + ..., what does ln(tan x/x)/x behave like?
   A. 1/3
   B. x^2/3
   C. x/3, which tends to 0
   D. 3/x
   Answer: C. ln(tan x/x) is about x^2/3; dividing by x gives x/3, tending to 0. (Key point: Series check)

4. If ln L = 0, what is L?
   A. e
   B. 0
   C. Does not exist
   D. 1
   Answer: D. L = e^0 = 1. (Key point: Final value)

## Related in Mathematics I

- [Limit of (tan x / x)^(1/x²)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Alimit-tanx-over-x-power-1-over-x2) — Mathematics I, chapter 1, asked 2×
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