---
title: "Limit with xeˣ and log(1+x)"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:limit-xe-x-minus-log"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/limit-xe-x-minus-log
site: BCT Engineering
---

# Limit with xeˣ and log(1+x)

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2071 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-xe-x-minus-log
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Evaluate lim_{x→0} (xeˣ − (1+x)log(1+x))/x².

## Short answer

By applying L’Hospital’s rule twice, the numerator’s second derivative at 0 is 1 and denominator’s is 2, giving limit 1/2. Series gives difference x²/2 + …

## Answer

Let L = lim_{x→0} [x eˣ − (1 + x) log(1 + x)]/x². Direct substitution gives 0/0. Apply L'Hospital's rule.

Derivative of the numerator: eˣ + x eˣ − log(1 + x) − 1. At x = 0 this is 1 + 0 − 0 − 1 = 0, and the denominator derivative is 2x → 0, so again 0/0.

```text
second derivative of numerator:
  d/dx [eˣ + x eˣ − log(1+x) − 1] = 2eˣ + x eˣ − 1/(1+x)
second derivative of denominator: 2
```
At x = 0: 2 + 0 − 1 = 1, so L = 1/2.

(Series check: x eˣ = x + x² + ...; (1+x) log(1+x) = x + x²/2 + ...; difference = x²/2 + ...)

**Answer: 1/2.**

## Check yourself

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

1. At x = 0 the limit has form:
   A. infinity - infinity
   B. 0/0
   C. infinity/infinity
   D. 1^infinity
   Answer: B. Both vanish. (Key point: Form)

2. After one application, why must we apply the rule again?
   A. The limit is infinite
   B. The new numerator and denominator still vanish at 0
   C. It is required
   D. The first answer was wrong
   Answer: B. Numerator derivative is 0 at 0, denominator 2x is 0. (Key point: Repeat)

3. Second derivative of the numerator at 0 is:
   A. -1
   B. 0
   C. 1
   D. 2
   Answer: C. 2 + 0 - 1 = 1. (Key point: Second derivative)

4. L =
   A. 2
   B. 0
   C. 1/2
   D. 1
   Answer: C. 1/2. (Key point: Result)

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