JEE (Main) Mathematics - LIMIT, CONTINUITY AND DIFFERENTIABILITY: Multiple Choice Questions (MCQs)

61. Which of the following is a real-valued function?
    A) \( f(x) = \sin(x) \)
    B) \( f(x) = \frac{1}{x-1} \)
    C) \( f(x) = \log(x) \)
    D) All of the above
    Answer: D) All of the above
62. Which of the following is not an algebraic function?
    A) Polynomial function
    B) Rational function
    C) Logarithmic function
    D) Exponential function
    Answer: C) Logarithmic function
63. The function \( f(x) = \frac{x^2 + 3x + 2}{x-1} \) is:
    A) Continuous at \( x = 1 \)
    B) Discontinuous at \( x = 1 \)
    C) Differentiable at \( x = 1 \)
    D) Neither continuous nor differentiable at \( x = 1 \)
    Answer: B) Discontinuous at \( x = 1 \)
64. The limit \( \lim_{x \to 0} \frac{\sin(x)}{x} \) is:
    A) 0
    B) 1
    C) Infinity
    D) Does not exist
    Answer: B) 1
65. If \( f(x) = x^3 - 3x^2 + 4 \), then the derivative of \( f(x) \) is:
    A) \( 3x^2 - 6x \)
    B) \( 3x^2 - 3x + 4 \)
    C) \( 6x - 6 \)
    D) \( 3x^2 + 6x \)
    Answer: A) \( 3x^2 - 6x \)
66. The derivative of \( f(x) = \sin(x) \) is:
    A) \( \cos(x) \)
    B) \( -\cos(x) \)
    C) \( -\sin(x) \)
    D) \( \sin(x) \)
    Answer: A) \( \cos(x) \)
67. The derivative of \( f(x) = \ln(x) \) is:
    A) \( \frac{1}{x} \)
    B) \( x \)
    C) \( x^2 \)
    D) \( \ln(x) \)
    Answer: A) \( \frac{1}{x} \)
68. The derivative of \( f(x) = e^x \) is:
    A) \( e^x \)
    B) \( e^{-x} \)
    C) \( x \cdot e^x \)
    D) \( \frac{1}{x} \)
    Answer: A) \( e^x \)
69. If \( f(x) = x^2 + 3x + 2 \) and \( g(x) = x^3 - 2x \), then \( \frac{d}{dx}(f(x) \cdot g(x)) \) is:
    A) \( 2x + 3 \)
    B) \( 3x^2 + 2x \)
    C) \( f'(x)g(x) + f(x)g'(x) \)
    D) \( x^2 + 2x \)
    Answer: C) \( f'(x)g(x) + f(x)g'(x) \)
70. The limit of \( \lim_{x \to \infty} \frac{1}{x^2} \) is:
    A) 1
    B) 0
    C) Infinity
    D) Does not exist
    Answer: B) 0
71. Which of the following is the derivative of the function \( f(x) = \tan(x) \)?
    A) \( \sec^2(x) \)
    B) \( \cos^2(x) \)
    C) \( \sec(x) \)
    D) \( -\sin(x) \)
    Answer: A) \( \sec^2(x) \)
72. If \( f(x) = \sin(x) \) and \( g(x) = \cos(x) \), then the derivative of \( f(x) \cdot g(x) \) is:
    A) \( \cos(x) \cdot \sin(x) \)
    B) \( \cos^2(x) - \sin^2(x) \)
    C) \( \cos(x) \cdot (-\sin(x)) + \sin(x) \cdot \cos(x) \)
    D) \( 0 \)
    Answer: C) \( \cos(x) \cdot (-\sin(x)) + \sin(x) \cdot \cos(x) \)
73. The derivative of the inverse function \( f^{-1}(x) \) is given by:
    A) \( \frac{1}{f'(x)} \)
    B) \( \frac{1}{f'(f^{-1}(x))} \)
    C) \( f'(x) \)
    D) None of the above
    Answer: B) \( \frac{1}{f'(f^{-1}(x))} \)
74. The maximum or minimum of the function \( f(x) = x^3 - 3x^2 + 3x + 1 \) occurs at:
    A) \( x = 0 \)
    B) \( x = 1 \)
    C) \( x = 2 \)
    D) \( x = -1 \)
    Answer: B) \( x = 1 \)
75. If the function \( f(x) \) is increasing on the interval \( (a, b) \), then:
    A) \( f'(x) > 0 \) for all \( x \in (a, b) \)
    B) \( f'(x) < 0 \) for all \( x \in (a, b) \)
    C) \( f'(x) = 0 \) for all \( x \in (a, b) \)
    D) \( f'(x) \) changes sign on \( (a, b) \)
    Answer: A) \( f'(x) > 0 \) for all \( x \in (a, b) \)

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