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

# Limit of (1/x²)^(tan x)

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2073 Chaitra
- Video: [LIMITS | INDETERMINATE FORM | LIM X tends 0 (1/x^2)^(tan x) | #maths #indeterminate_forms](https://www.youtube.com/watch?v=kAusD83Ddyg) — MaTH LEARNS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/limit-1-over-x2-power-tan-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Evaluate lim_{x→0} (1/x²)^(tan x).

## Short answer

Limit is of the form ∞^0. Take logarithm: ln L = -2 lim tan x ln|x|. Apply L'Hospital's rule to get ln L = 0, so L = e^0 = 1.

## Answer

Let L = lim_{x→0} (1/x²)^(tan x). As x → 0, 1/x² → ∞ and tan x → 0, so this is the form ∞⁰.

Take logarithms:

ln L = lim tan x · ln(1/x²) = −2 lim tan x · ln|x|

First let x → 0⁺ (so |x| = x):

ln L = −2 lim (ln x)/(cot x)  (form ∞/∞)

Apply L'Hospital's rule:

```text
ln L = −2 lim (1/x)/(−cosec²x)
     = 2 lim sin²x / x
     = 2 lim (sin x/x) · sin x
     = 2 · 1 · 0 = 0
```
As x → 0⁻, tan x · ln(1/x²) only changes sign (tan x is odd and ln(1/x²) is even), so it also tends to 0. Hence ln L = 0 from both sides.

**Answer: L = e⁰ = 1.**

## Check yourself

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

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

2. ln L becomes:
   A. -2 lim (ln x)/(cot x)
   B. -2 lim ln x/tan x
   C. 2 lim ln x tan x
   D. lim x/tan x
   Answer: A. tan x ln(1/x^2) = -2 tan x ln x = -2 ln x/cot x. (Key point: Rewrite)

3. The expression becomes:
   A. lim 1/x
   B. lim cosec^2 x
   C. 2 lim sin^2 x / x
   D. -2 lim x
   Answer: C. (1/x)/(-cosec^2 x) times -2. (Key point: After L'Hospital)

4. L =
   A. infinity
   B. e
   C. 1
   D. 0
   Answer: C. ln L = 0. (Key point: Result)

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