JEE (Main) Mathematics - LIMIT, CONTINUITY AND DIFFERENTIABILITY Multiple Choice Questions (MCQs)
JEE (Main) Mathematics - LIMIT, CONTINUITY AND DIFFERENTIABILITY MCQs
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Which of the following is the domain of the function \( f(x) = \log(x-3) \)?
- A) \( x > 3 \)
- B) \( x \geq 3 \)
- C) \( x < 3 \)
- D) \( x \neq 3 \)
Answer: A) \( x > 3 \)
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The function \( f(x) = \frac{1}{x^2 - 1} \) is continuous for:
- A) \( x = 1 \)
- B) \( x = -1 \)
- C) \( x \neq 1, -1 \)
- D) All real values of \( x \)
Answer: C) \( x \neq 1, -1 \)
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The limit \( \lim_{x \to 0} \frac{\sin x}{x} \) is:
- A) 0
- B) 1
- C) Does not exist
- D) \( \infty \)
Answer: B) 1
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Which of the following is the inverse function of \( f(x) = 2x + 3 \)?
- A) \( f^{-1}(x) = \frac{x-3}{2} \)
- B) \( f^{-1}(x) = \frac{x+3}{2} \)
- C) \( f^{-1}(x) = x - 3 \)
- D) \( f^{-1}(x) = 2x - 3 \)
Answer: A) \( f^{-1}(x) = \frac{x-3}{2} \)
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The function \( f(x) = \frac{1}{x} \) is differentiable at:
- A) \( x = 0 \)
- B) \( x \neq 0 \)
- C) \( x = 1 \)
- D) \( x = -1 \)
Answer: B) \( x \neq 0 \)
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For the function \( f(x) = \log x \), the derivative is:
- A) \( \frac{1}{x} \)
- B) \( \frac{1}{x^2} \)
- C) \( \ln x \)
- D) \( x \ln x \)
Answer: A) \( \frac{1}{x} \)
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Which of the following functions is continuous for all real values of \( x \)?
- A) \( \frac{1}{x-2} \)
- B) \( \sin x \)
- C) \( \log x \)
- D) \( \frac{1}{x} \)
Answer: B) \( \sin x \)
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The function \( f(x) = x^3 - 5x + 7 \) is:
- A) Continuous, but not differentiable
- B) Differentiable, but not continuous
- C) Both continuous and differentiable
- D) Neither continuous nor differentiable
Answer: C) Both continuous and differentiable
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The function \( f(x) = \sqrt{x} \) is differentiable for:
- A) \( x > 0 \)
- B) \( x \geq 0 \)
- C) \( x \neq 0 \)
- D) None of the above
Answer: A) \( x > 0 \)
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Which of the following functions is a polynomial function?
- A) \( \frac{1}{x^2 + 1} \)
- B) \( x^2 - 5x + 3 \)
- C) \( \log(x+2) \)
- D) \( \sin x \)
Answer: B) \( x^2 - 5x + 3 \)
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The function \( f(x) = e^x \) is:
- A) Continuous and differentiable for all real values of \( x \)
- B) Continuous, but not differentiable
- C) Neither continuous nor differentiable
- D) Differentiable, but not continuous
Answer: A) Continuous and differentiable for all real values of \( x \)
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Which of the following functions is not a rational function?
- A) \( \frac{1}{x^2 + 1} \)
- B) \( \frac{x^2 - 1}{x+2} \)
- C) \( \frac{x^3}{x^2 + 2x} \)
- D) \( \log(x) \)
Answer: D) \( \log(x) \)
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For the function \( f(x) = \tan^{-1}x \), the derivative is:
- A) \( \frac{1}{1+x^2} \)
- B) \( \frac{1}{x^2+1} \)
- C) \( \frac{x}{1+x^2} \)
- D) \( 2x \)
Answer: A) \( \frac{1}{1+x^2} \)
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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
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Which of the following functions is the inverse of \( f(x) = \sin x \), for \( x \in [0, \pi] \)?
- A) \( \sin^{-1}x \)
- B) \( \cos^{-1}x \)
- C) \( \tan^{-1}x \)
- D) \( \sec^{-1}x \)
Answer: A) \( \sin^{-1}x \)
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