Q.7 The value of \(\lim_{n\to\infty} \frac{3n^2 + 5n + 4}{4 + 2n^2}\) is

Introduction

Finding the limit of rational functions like (3n²+5n+4)/(4+2n²) as n tends to infinity is a standard concept in calculus and sequences, and it often appears in competitive exams as a quick MCQ. Understanding how to simplify such expressions using leading coefficients helps solve these questions accurately and efficiently.

Step-by-step solution of the limit

Consider L = lim_n→∞(3n²+5n+4)/(4+2n²).

Identify the highest power of n:

• In the numerator, the highest power is n² with coefficient 3.
• In the denominator, the highest power is n² with coefficient 2.

Divide numerator and denominator by n²:

(3n²+5n+4)/(4+2n²) = (3 + 5/n + 4/n²)/(4/n² + 2).

Use limits of the smaller terms:

lim_n→∞5/n = 0 and lim_n→∞4/n² = 0.

lim_n→∞4/n² = 0 in the denominator as well.

Evaluate the limit:

L = 3 + 0 + 0 / 0 + 2 = 3/2 = 1.5.

So the value of the limit is 1.5.

Explanation of each option

The MCQ options are:

(A) 0 (B) 0.75 (C) 1.5 (D) 3.

Option (A) 0

A limit of 0 would occur if the degree of the numerator were less than the degree of the denominator, making the fraction vanish as n grows large.

Here, both numerator and denominator have the same degree (2), so the limit is a non-zero ratio of leading coefficients, not 0.

Therefore, option (A) 0 is incorrect.

Option (B) 0.75

0.75 = 3/4, which might come from incorrectly taking coefficients 3 and 4 instead of the correct leading coefficients 3 and 2.

The constant 4 in the denominator is not the leading term; the leading term is 2n², so 3/4 is based on a misunderstanding of which terms dominate for large n.

Hence, option (B) 0.75 is incorrect.

Option (C) 1.5

Using the rule for rational functions with equal degrees, the limit as n→∞ is the ratio of leading coefficients: 3/2 = 1.5.

This matches the computed limit L = 1.5, so option (C) 1.5 is correct.

Option (D) 3

A limit of 3 would arise if the denominator’s leading coefficient were 1 (giving 3/1 = 3) or if the denominator grew slower than the numerator, which is not the case.

Since the denominator has leading coefficient 2 and same degree as the numerator, the correct ratio is 3/2, not 3.

Therefore, option (D) 3 is incorrect.

Key takeaways for similar limits

When numerator and denominator have the same degree, the limit at infinity of a rational function is the ratio of the leading coefficients.

When the numerator’s degree is less, the limit is 0; when it is greater, the limit diverges to ∞ or −∞ depending on signs.

Recognizing and applying this pattern quickly is crucial for solving MCQs on limits in competitive exams.