JEE (Main) Mathematics - Limit, Continuity and Differentiability MCQs

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JEE (Main) Mathematics - LIMIT, CONTINUITY AND DIFFERENTIABILITY MCQs

Which of the following is the graph of the function \( f(x) = |x| \)?
A) A straight line passing through the origin.
B) A V-shaped curve opening upwards.
C) A parabola opening upwards.
D) A straight line with a slope of 1.
Answer: B) A V-shaped curve opening upwards.
The limit \( \lim_{x \to 0} \frac{\sin x}{x} \) is equal to:
A) 0
B) 1
C) Does not exist
D) \( \infty \)
Answer: B) 1
Which of the following is true for the function \( f(x) = x^2 \)?
A) The function is not differentiable at \( x = 0 \).
B) The function is continuous but not differentiable at \( x = 0 \).
C) The function is both continuous and differentiable for all real values of \( x \).
D) The function is neither continuous nor differentiable.
Answer: C) The function is both continuous and differentiable for all real values of \( x \).
If \( f(x) = x^3 - 3x + 2 \), what is the derivative of the function?
A) \( 3x^2 - 3 \)
B) \( 3x^2 - 6x \)
C) \( 3x^2 - 3x + 2 \)
D) \( 3x^2 + 3 \)
Answer: A) \( 3x^2 - 3 \)
The function \( f(x) = \frac{1}{x} \) is differentiable at:
A) \( x = 0 \)
B) \( x = 1 \)
C) \( x = -1 \)
D) \( x \neq 0 \)
Answer: D) \( x \neq 0 \)
The limit \( \lim_{x \to \infty} \frac{1}{x} \) is:
A) 0
B) 1
C) Does not exist
D) \( \infty \)
Answer: A) 0
Which of the following is true for the function \( f(x) = x^2 - 4x + 3 \)?
A) The function has a turning point at \( x = 2 \).
B) The function is not continuous at \( x = 2 \).
C) The function is both continuous and differentiable for all real values of \( x \).
D) The function is neither continuous nor differentiable.
Answer: C) The function is both continuous and differentiable for all real values of \( x \).
The derivative of the sum of two functions \( f(x) \) and \( g(x) \) is:
A) \( f'(x) + g'(x) \)
B) \( f(x) + g(x) \)
C) \( f'(x) \cdot g'(x) \)
D) \( f(x) \cdot g(x) \)
Answer: A) \( f'(x) + g'(x) \)
Which of the following statements is true for the function \( f(x) = \log(x) \)?
A) The function is continuous and differentiable for all real values of \( x \).
B) The function is continuous but not differentiable at \( x = 0 \).
C) The function is not continuous at \( x = 0 \).
D) The function is both continuous and differentiable for all \( x > 0 \).
Answer: D) The function is both continuous and differentiable for all \( x > 0 \).
The limit \( \lim_{x \to 0} \frac{e^x - 1}{x} \) is:
A) 0
B) 1
C) Does not exist
D) \( \infty \)
Answer: B) 1
The graph of \( y = \sin x \) is:
A) A straight line passing through the origin.
B) A periodic wave-like curve with amplitude 1.
C) A U-shaped curve.
D) A straight line with a slope of 1.
Answer: B) A periodic wave-like curve with amplitude 1.
The derivative of the product of two functions \( f(x) \) and \( g(x) \) is:
A) \( f'(x) \cdot g(x) + f(x) \cdot g'(x) \)
B) \( f'(x) + g'(x) \)
C) \( f(x) \cdot g(x) \)
D) \( f'(x) \cdot g'(x) \)
Answer: A) \( f'(x) \cdot g(x) + f(x) \cdot g'(x) \)
If \( f(x) = \tan x \), then the derivative of the function is:
A) \( \sec^2 x \)
B) \( \cos x \)
C) \( \sin x \)
D) \( \cot x \)
Answer: A) \( \sec^2 x \)
Which of the following functions is continuous and differentiable at \( x = 0 \)?
A) \( f(x) = \sqrt{x} \)
B) \( f(x) = \frac{1}{x} \)
C) \( f(x) = e^x \)
D) \( f(x) = \log(x) \)
Answer: C) \( f(x) = e^x \)
For the function \( f(x) = x^3 - 6x^2 + 9x \), what is the value of \( f'(2) \)?
A) 0
B) -6
C) 6
D) 12
Answer: A) 0

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JEE (Main) Mathematics - LIMIT, CONTINUITY AND DIFFERENTIABILITY MCQs

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