GATE 2025 DA Q.32: solution and answer

GATE 2025 · DAQ.32 · 1 mark · NAT

$t \to + \infty lim t^{2} + t - t =$ (Round off to one decimal place)

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Answer and solution

ANSWER

The limit equals $0.5$ .

Official answer: 0.5Source: Official answer key, GATE 2025 DA (IIT Roorkee)

The limit has the form $\infty - \infty$ , so rationalise by multiplying and dividing by the conjugate $t^{2} + t + t$ .
$t^{2} + t - t = \frac{( t ^{2} + t ) - t ^{2}}{t ^{2} + t + t} = \frac{t}{t ^{2} + t + t}$
For $t > 0$ we have $t^{2} + t = t 1 + \frac{1}{t}$ . Divide the numerator and denominator by $t$ : $\frac{t}{t ^{2} + t + t} = \frac{1}{1 + \frac{1}{t} + 1}$
As $t \to + \infty$ , $\frac{1}{t} \to 0$ , so $1 + \frac{1}{t} \to 1$ .
$t \to + \infty lim \frac{1}{1 + \frac{1}{t} + 1} = \frac{1}{1 + 1} = \frac{1}{2} = 0.5$

KEY IDEA

A difference of two large square-root terms ( $\infty - \infty$ ) is resolved by multiplying by the conjugate, which turns it into a ratio whose limit can be read off after dividing by $t$ .

WATCH OUT FOR

Substituting $t^{2} + t \approx t$ and concluding the limit is $0$ . The neglected term $\frac{1}{2}$ is exactly the answer, since $t^{2} + t \approx t + \frac{1}{2}$ for large $t$ .

From the GATE 2025 DA paper. Every question of this paper · Download the full paper · All 6 Limits & continuity questions, with notes

GATE 2025 DA Q.32: solution and answer

Answer and solution

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