Limits

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The value of lim(x→∞) (3x² + 2x + 1)/(2x² - x + 3) is

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3/2

2/3

∞

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About This Question

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

Solution

Correct Answer:

3/2

Dominant Term Analysis: In rational functions where $x \to \infty$ , the behavior is determined by the highest powers of $x$ .
Divide by Highest Power: Divide numerator and denominator by $x^{2}$ :
- $lim_{x \to \infty} \frac{3 + 2/ x + 1/ x ^{2}}{2 - 1/ x + 3/ x ^{2}}$
Evaluate Terms: As $x \to \infty$ , terms like $2/ x, 1/ x^{2}$ etc. approach zero.
- $\frac{3 + 0 + 0}{2 - 0 + 0} = \frac{3}{2}$ .

General Rule: If degrees of numerator and denominator are equal, the limit is the ratio of their leading coefficients.

This medium 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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Solution

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