Continuity and Differentiability – important questions for Boards 2027
CBSE Class 12 Maths
Determine the open interval of x-values on which the derivative of sin⁻¹x is defined.
Answer
∴ x ∈ (−1, 1)
If g is continuous at c, which statement must be true for every real constant λ? (A) λg is continuous at c (B) λg is continuous only if λ>0 (C) λg is discontinuous when λ=−1 (D) λg is continuous only if g(c)=0
Answer
(A) λg is continuous at c — a constant multiple of a continuous function is continuous.
Evaluate lim₍x→π/6₎ sin x and compare it with sin(π/6) to verify continuity of sine at x=π/6.
Answer
lim x→π/6 sin x = sin(π/6) = 1/2 ∴ lim x→π/6 sin x = sin(π/6), so sine is continuous at x = π/6.
By putting y = sin⁻¹x and differentiating x = sin y, derive dy/dx in terms of x for −1 < x < 1.
Answer
Let y = sin⁻¹x, so x = sin y. 1 = cos y · dy/dx cos² y = 1 − sin² y = 1 − x² Since −1 < x < 1, cos y > 0. ∴ dy/dx = 1/√(1 − x²)
For f(x)=1/(x²+1), find f(0) and evaluate limₓ→₀f(x). Hence verify continuity at x=0 using reciprocal continuity.
Answer
f(0) = 1/(0² + 1) = 1 lim x→0 f(x) = lim x→0 1/(x² + 1) = 1/(0² + 1) = 1 ∴ lim x→0 f(x) = f(0), so f is continuous at x = 0.
If f(x) = sin((2x + 1)³), write it as a composite of three functions and find df/dx using the successive derivatives.
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Prove that f(x) = sin(x²) is continuous for every real x. Also prove that f(x) = |1 − x + |x|| is continuous for every real x, using the continuity of the modulus function and the rules for sums and compositions of continuous functions.
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Find d(log x)/dx for x > 0.
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Find the real value of x at which f(x)=(x+1)/(x²−4) is not continuous.
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Evaluate limₓ→₁ sin(x²) and hence verify that sin(x²) is continuous at x=1.
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42 more probable questions in this chapter, every one with a full answer.
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