Indian Navy AA (Artificer Apprentice) Limits and Derivatives — practice questions

Explore 2 free Limits and Derivatives questions in Indian Navy AA (Artificer Apprentice) Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Limits and Derivatives questions and exercises with answers

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A function f is continuous at x = a if and only if:Read the answer and worked solution →The TANGENT line to y = x² at the point (2, 4) has equation:Read the answer and worked solution →

Worked practice questions

A function f is continuous at x = a if and only if:

lim_{x→a} f(x) exists AND equals f(a)
f(a) is finite (continuity just means defined)
f is differentiable at a (this is stronger)
lim_{x→a-} f(x) exists (only LHL needed)

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Correct answer: A. lim_{x→a} f(x) exists AND equals f(a)

**Continuity at a requires:** (1) f(a) defined; (2) lim_{x→a} f(x) exists (LHL = RHL) (3) lim equals f(a). All three must hold. Differentiability implies continuity, but not vice versa.

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The TANGENT line to y = x² at the point (2, 4) has equation:

$y = 4x$ (slope through origin)
$y = 4x + 4$ (wrong y-intercept)
$y = x − 2$ (incorrect calculation)
$y = 4x − 4$ (correct tangent line)

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Correct answer: D. $y = 4x − 4$ (correct tangent line)

dy/dx = 2x. At x = 2: slope = 4. Tangent: y − 4 = 4(x − 2) → y = 4x − 4.