The limit p = x (x2 + x + 22x - + 2 x) has a finite value for a real . The value of and the corresponding limit p are

$\alpha = -2\pi$, and $p = 2\pi$

$\alpha = 2\pi$, and $p = 3\pi$

GATE 2022 ME (ME1) – Question 11

Engineering Mathematics · Calculus: Limit, continuity, differentiability, mean value theorems and indeterminate forms · 1 mark · Multiple choice

The limit $p = \lim_{x \to \pi}\left(\frac{x^2 + \alpha x + 2\pi^2}{x - \pi + 2\sin x}\right)$ has a finite value for a real $\alpha$. The value of $\alpha$ and the corresponding limit $p$ are

$\alpha = -3\pi$, and $p = \pi$
$\alpha = -2\pi$, and $p = 2\pi$
$\alpha = \pi$, and $p = \pi$
$\alpha = 2\pi$, and $p = 3\pi$

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Correct answer: (A) $\alpha = -3\pi$, and $p = \pi$

Explanation

At $x = \pi$ the denominator is zero, so the numerator must also be zero: $\pi^2 + \alpha\pi + 2\pi^2 = 0$, giving $\alpha = -3\pi$. By L'Hopital's rule the limit is $\frac{2x + \alpha}{1 + 2\cos x}$ at $x = \pi$, which is $\frac{2\pi - 3\pi}{1 - 2} = \pi$.