Limits

Easymathematics

The value of lim(x→0) (sin x)/x is

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∞

Does not exist

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Subject: mathematics
Chapter: limit, continuity and differentiability
Topic: limits
Difficulty: Easy
Year: 2025

Solution

Correct Answer:

Indeterminate Form: As $x \to 0$ , $sin x \to 0$ , resulting in $0/0$ .
Standard Limit: This is a fundamental trigonometric limit derived using Squeeze Theorem or Taylor series expansion.
- $sin x = x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots$
- $\frac{s i n x}{x} = 1 - \frac{x ^{2}}{3 !} + \frac{x ^{4}}{5 !} - \dots$
Result: As $x \to 0$ , all terms with $x$ vanish, leaving $lim_{x \to 0} \frac{s i n x}{x} = 1$ .

This easy difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of limits. It appeared in the 2025 exam.

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Limits

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Limit $n \to \infty$ of $\frac{1}{n} [(n + 1) (n + 2) ... (n + n)]^{1/ n}$ is:

Let $f (x)$ be a polynomial of degree 4 having extreme values at $x = 1$ and $x = 2$ . If $\lim_{x \to 0} ...

Evaluate the limit: $L = lim_{x \to 0} \frac{l n ( 1 + x + x ^{2} ) + l n ( 1 - x + x ^{2} )}{s e c x - c o s x}$