---
title: "Find a so that limit is finite"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:limit-a-sinx-minus-sin2x-find-a"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/limit-a-sinx-minus-sin2x-find-a
site: BCT Engineering
---

# Find a so that limit is finite

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2069 Chaitra
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/limit-a-sinx-minus-sin2x-find-a
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

If lim_{x→0} (a sin x − sin 2x)/tan³x is finite, find the value of a and the limit.

## Short answer

Expanding numerator: (a−2)x + (−a/6 + 4/3)x³ + … For a finite limit, a−2 = 0, so a = 2. Then the x³ coefficient gives limit 1.

## Answer

Let L = lim_{x→0} (a sin x − sin 2x)/tan³x. The denominator tan³x → 0 like x³, so for the limit to be finite the numerator must also vanish to the same order. Use series:

```text
a sin x   = a(x − x³/6 + ...)
sin 2x    = 2x − (2x)³/6 + ... = 2x − 4x³/3 + ...
numerator = (a − 2)x + (−a/6 + 4/3)x³ + ...
tan³x     = x³ + ...
```
Dividing by x³, the term (a − 2)/x² blows up unless a − 2 = 0. So **a = 2**.

With a = 2 the x³ coefficient of the numerator is −2/6 + 4/3 = −1/3 + 4/3 = 1, so

L = lim (x³ + ...)/(x³ + ...) = 1.

(By L'Hospital's rule: the numerator derivative a cos x − 2 cos 2x must vanish at 0, giving a = 2. This confirms the value.)

**Answer: a = 2 and the limit is 1.**

## Check yourself

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

1. tan^3 x behaves like:
   A. x^4
   B. x^2
   C. x
   D. x^3
   Answer: D. tan x ~ x. (Key point: Order)

2. For finiteness, the x term of the numerator must vanish: a - 2 = 0 gives:
   A. a = 4
   B. a = -2
   C. a = 1
   D. a = 2
   Answer: D. Otherwise (a-2)/x^2 blows up. (Key point: Condition)

3. With a = 2, the x^3 coefficient is:
   A. -1
   B. 4/3
   C. 1
   D. 1/3
   Answer: C. -2/6 + 4/3 = 1. (Key point: x^3 coefficient)

4. The limit equals:
   A. 1/3
   B. 1
   C. 2
   D. 0
   Answer: B. x^3/x^3. (Key point: Limit)

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