If lim_x (1+a/x+bx^2)^2x=e^2, then the values of a and b are:

B. a=1, b∈R. Step 1: This is a 1^ form, so the limit equals e^L with L=lim_x 2x(a/x+bx^2). Step 2: L=lim_x (2a+2b/x)=2a, so the limit is e^2a. Step 3: e^2a=e^2 gives a=1. The b-term vanishes, so b can be any real number. Check: Take a=1, b=0: (1+1/x)^2x e^2, so b need not be 2. Trap: (A) assumes b must also equal 2, but 2b/x 0 for every b.

Question 17 of 21 · Limits

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If $l im_{x \to \infty} (1 + \frac{a}{x} + \frac{b}{x ^{2}})^{2 x} = e^{2}$ , then the values of a and b are:

Answer (Detailed Solution Below)View answer ▾

Answer: Option B — $a = 1$ , $b \in R$

Detailed Solution

This is a $1^{\infty}$ form, so the limit equals $e^{L}$ with $L = l im_{x \to \infty} 2 x (\frac{a}{x} + \frac{b}{x ^{2}})$ .

$L = l im_{x \to \infty} (2 a + \frac{2 b}{x}) = 2 a$ , so the limit is $e^{2 a}$ .

$e^{2 a} = e^{2}$ gives $a = 1$ . The b-term vanishes, so b can be any real number.

✓ Check

Take

a = 1

b = 0

(1 + \frac{1}{x})^{2 x} \to e^{2}

, so b need not be 2.

⚠ Common trap

(A) assumes b must also equal 2, but

\frac{2 b}{x} \to 0

for every b.

Quick tip

$1^{\infty}$ form: limit $= e^{l i m 2 x (a / x + b / x^{2})} = e^{2 a} \Rightarrow a = 1$ , b is free

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Limits — Preparation

This question tests your understanding of Limits, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

The limit of a function describes the value the function approaches as its input approaches a given point. Limits are the foundation of calculus, because continuity, derivatives and integrals are all defined using them. This topic covers left-hand and right-hand limits, the algebra of limits, and standard methods of evaluation: direct substitution, factorisation, rationalisation, and dividing by the highest power for limits at infinity. It also covers the standard limits, such as (sin x)/x -> 1 as x -> 0, (e^x - 1)/x -> 1, (a^x - 1)/x -> log a, and (1 + x)^(1/x) -> e. Most exam questions are 0/0 forms that reduce to one of these after simplification. A limit exists only when the left-hand and right-hand limits are equal, which is especially important for the modulus and greatest integer functions. Practise the limits questions below, each with a step-by-step solution.

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If $l im_{x \to \infty} (1 + \frac{a}{x} + \frac{b}{x ^{2}})^{2 x} = e^{2}$ , then the values of a and b are: