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Questions › IIT JAM › IIT JAM 2021 › Q35
2 marksMathematics › Limits, continuity and differentiability › Differentiability and its link to continuityHardabout 2 min

IIT JAM 2021 Question 35: Limits, continuity and differentiability

Let fff be defined on (−π2,π2)\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)(−2π​,2π​) by f(x)=cos⁡(π2+∣x∣)sin⁡(π2−∣x∣)f(x) = \dfrac{\cos\left(\tfrac{\pi}{2} + |x|\right)}{\sin\left(\tfrac{\pi}{2} - |x|\right)}f(x)=sin(2π​−∣x∣)cos(2π​+∣x∣)​. Then,

Select all that apply.

Answer and worked solution

Answer: (B) and (C)

Using cos⁡(π2+∣x∣)=−sin⁡∣x∣\cos\left(\tfrac{\pi}{2} + |x|\right) = -\sin|x|cos(2π​+∣x∣)=−sin∣x∣ and sin⁡(π2−∣x∣)=cos⁡∣x∣\sin\left(\tfrac{\pi}{2} - |x|\right) = \cos|x|sin(2π​−∣x∣)=cos∣x∣, the function is f(x)=−tan⁡∣x∣f(x) = -\tan|x|f(x)=−tan∣x∣, which is continuous on the interval, so (A) is false.

The right-hand derivative at 000 is −1-1−1, matching (C), while the left-hand derivative is +1+1+1. Since the one-sided derivatives differ, fff is continuous but not differentiable at 000, so (B) holds and (D) is false (no two-sided f′(0)f'(0)f′(0) exists). The correct statements are (B) and (C).

The trap: (D) f′(0)=−1f'(0) = -1f′(0)=−1

The right-derivative equals −1-1−1, but the left-derivative is +1+1+1, so a single f′(0)f'(0)f′(0) does not exist; (C) is a one-sided limit while (D) wrongly claims a full derivative.

Plain-text version

Let f be defined on (-π/2, π/2) by f(x) = (cos (π/2 + |x|))/(sin (π/2 - |x|)). Then, (A) f is not continuous at x = 0 (B) f is continuous but not differentiable at x = 0 (C) lim_(x → 0⁺) (f(x) - f(0))/x = -1 (D) f′(0) = -1

Also written as: Let f be defined on (-pi/2, pi/2) by f(x) = (cos (pi/2 + |x|))/(sin (pi/2 - |x|)). Then,

Limits, continuity and differentiability has come up 122 times in 7 exams since 2006.

What this question tests

This question is on Differentiability and its link to continuity, part of Limits, continuity and differentiability in Mathematics.

When a derivative exists and why differentiability implies continuity but not conversely.

TestStop has 12 past-paper questions on this subtopic; the nearest in year are below.

  • IIT JAM 2022 · Q34Let f be defined by f(x) = |x| + |cos (π/2 - x)| for x ∈ (-π/2, π/2). Then
Take this paper as a timed mockMore on Differentiability and its link to continuity

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