Which of the following is NOT true?

Answer (Detailed Solution Below)View answer ▾

Answer: Option D — $l im_{x \to 0^{+}} \frac{cos x}{1 + 2 x} = 0$

Detailed Solution

(A) $e^{x}$ grows faster than $x$ , so $\frac{x}{e ^{x}} \to 0$ : true.

(B) Put $t = \frac{1}{x} \to \infty$ : $\frac{1}{x e ^{1/ x}} = \frac{t}{e ^{t}} \to 0$ : true. (C) $\frac{s in 0}{1 + 0} = 0$ : true.

(D) $\frac{cos x}{1 + 2 x} \to \frac{cos 0}{1} = 1 \neq = 0$ , so (D) is NOT true.

✓ Check

x = 0.001

\frac{cos x}{1 + 2 x} \approx 0.998

, close to 1, not 0.

⚠ Common trap

(B) looks like a

\frac{1}{0}

form, but

e^{1/ x}

grows much faster than

x

shrinks, so its limit really is 0.

Quick tip

Direct substitution: $\frac{s i n 0}{1} = 0$ but $\frac{c o s 0}{1} = 1$

💬 Discussions on this question

Ask a doubt ›

Still stuck on this question? Ask our faculty and fellow students — answers appear here.

Ask a doubt

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.

‹ All Limits questions Browse all topics ›

Which of the following is NOT true?

💬 Discussions on this question

More Limits Questions

More Quantitative Aptitude Questions

Limits — Preparation