JEE Main - Limits and continuity Online Test (MCQ Quiz)

1. What is the fundamental concept of a limit in calculus?

A) The value a function attains at a specific point. B) The behavior of a function as its input approaches a certain value. C) The area under the curve of a function. D) The slope of the tangent line to a function.

2. For a limit to exist at a point 'c', what condition must be met?

A) The function must be defined at 'c'. B) The left-hand limit and the right-hand limit must be equal. C) The function must be continuous at 'c'. D) The function must be differentiable at 'c'.

3. If lim (x->a) f(x) = L, what does this imply?

A) f(a) = L B) f(x) is continuous at x=a C) As x gets arbitrarily close to 'a', f(x) gets arbitrarily close to L D) The derivative of f(x) at x=a is L

4. Which of the following is NOT a condition for a function f(x) to be continuous at a point 'c'?

A) f(c) must be defined. B) The limit of f(x) as x approaches 'c' must exist. C) The limit of f(x) as x approaches 'c' must be equal to f(c). D) The function must be differentiable at 'c'.

5. What is the value of lim (x->0) (sin(x)/x)?

A) 0 B) 1 C) Undefined D) Infinity

6. Evaluate lim (x->2) (x^2 - 4)/(x - 2).

A) 0 B) 2 C) 4 D) Undefined

7. If lim (x->a) f(x) = L and lim (x->a) g(x) = M, then what is lim (x->a) [f(x) + g(x)]?

A) L + M B) L - M C) L * M D) L / M

8. The limit of a constant function f(x) = k as x approaches any value 'a' is:

A) x B) a C) k D) 0

9. What type of discontinuity occurs when lim (x->c) f(x) exists, but f(c) is not defined or f(c) is not equal to the limit?

A) Removable discontinuity B) Jump discontinuity C) Infinite discontinuity D) Oscillating discontinuity

10. Consider the function f(x) = |x|/x for x != 0. What is lim (x->0+) f(x)?

A) -1 B) 0 C) 1 D) Undefined

11. Consider the function f(x) = |x|/x for x != 0. What is lim (x->0-) f(x)?

A) -1 B) 0 C) 1 D) Undefined

12. For the function f(x) = |x|/x, does the limit exist as x approaches 0?

A) Yes, it is 0. B) Yes, it is 1. C) No, because the left-hand limit and right-hand limit are not equal. D) No, because the function is not defined at 0.

13. If a function f(x) is continuous on a closed interval [a, b], what property does it possess according to the Intermediate Value Theorem?

A) It attains a maximum and minimum value on the interval. B) It takes on every value between f(a) and f(b). C) It is differentiable on the interval. D) Its derivative is constant on the interval.

14. What is the value of lim (x->0) (e^x - 1)/x?

A) 0 B) 1 C) e D) Undefined

15. Evaluate lim (x->infinity) (1/x).

A) Infinity B) 1 C) 0 D) Undefined

16. What is the limit of x^n as x approaches infinity, where n > 0?

A) 0 B) 1 C) Infinity D) Depends on n

17. If lim (x->a) f(x) = L, what is lim (x->a) [c * f(x)], where 'c' is a constant?

A) c * L B) L / c C) c + L D) c - L

18. What is the left-hand limit of f(x) = x^2 at x=3?

A) 3 B) 6 C) 9 D) Undefined

19. What is the right-hand limit of f(x) = x^2 at x=3?

A) 3 B) 6 C) 9 D) Undefined

20. Is the function f(x) = x^2 continuous at x=3?

A) Yes B) No C) Only if f(3) is defined D) Only if the derivative exists

21. Which theorem states that if a function is continuous on a closed interval [a, b], then it attains its maximum and minimum values on that interval?

A) Intermediate Value Theorem B) Mean Value Theorem C) Extreme Value Theorem D) Bolzano's Theorem

22. Evaluate lim (x->pi) sin(x).

A) -1 B) 0 C) 1 D) pi

23. What is lim (x->0) cos(x)?

A) 0 B) 1 C) Undefined D) pi/2

24. Consider the function f(x) = 1/x. What is lim (x->0) f(x)?

A) 0 B) 1 C) Infinity D) Does not exist

25. What is lim (x->0) (tan(x)/x)?

A) 0 B) 1 C) Infinity D) Undefined

26. If lim (x->a) f(x) = L and lim (x->a) g(x) = M, then what is lim (x->a) [f(x) * g(x)]?

A) L * M B) L + M C) L - M D) L / M

27. What is the limit of a polynomial function P(x) as x approaches 'a'?

A) P'(a) B) P(a) C) 0 D) Infinity

28. If lim (x->a) f(x) = L and lim (x->a) g(x) = M (where M != 0), then what is lim (x->a) [f(x) / g(x)]?

A) L / M B) M / L C) L * M D) L + M

29. What is a 'jump discontinuity'?

A) The left-hand limit and right-hand limit are finite but unequal. B) The limit does not exist because the function goes to infinity. C) The function is undefined at the point. D) The left-hand limit equals the right-hand limit, but not f(c).

30. Consider f(x) = {x if x is rational, 0 if x is irrational}. What is lim (x->0) f(x)?

A) 0 B) 1 C) Undefined D) Depends on whether 0 is rational

31. Is the function f(x) = sin(1/x) continuous at x=0?

A) Yes, because sin(1/0) can be evaluated. B) No, because 1/x is not defined at x=0. C) No, because the limit as x approaches 0 does not exist (oscillating discontinuity). D) Yes, because sin(x) is continuous.

32. What is the value of lim (x->infinity) (sin(x)/x)?

A) 1 B) 0 C) Infinity D) Undefined

33. If f(x) is continuous at x=c, then lim (x->c) f(x) = ?

A) f'(c) B) f(c) C) 0 D) Undefined

34. Evaluate lim (x->1) (x^3 - 1)/(x - 1).

A) 0 B) 1 C) 3 D) Infinity

35. What is the limit of the function f(x) = 2x + 3 as x approaches 5?

A) 2 B) 3 C) 5 D) 13

36. What is lim (x->0) (1 - cos(x))/x?

A) 0 B) 1 C) 1/2 D) Undefined

37. If lim (x->a) f(x) exists, and lim (x->a) g(x) exists, does lim (x->a) [f(x) / g(x)] necessarily exist?

A) Yes B) No, unless g(a) != 0 C) No, unless lim (x->a) g(x) != 0 D) No, unless f(x) and g(x) are continuous

38. What does it mean for a function to have an infinite discontinuity at x=c?

A) lim (x->c+) f(x) = infinity and lim (x->c-) f(x) = infinity B) lim (x->c+) f(x) = -infinity and lim (x->c-) f(x) = -infinity C) Either lim (x->c+) f(x) or lim (x->c-) f(x) is infinity or -infinity D) The function is undefined at x=c.

39. Consider the function f(x) = {x^2 if x < 0, x if x >= 0}. Is f(x) continuous at x=0?

A) Yes B) No, because the function definition changes. C) No, because the left-hand limit is 0 and the right-hand limit is 0, but f(0)=0. D) No, because the derivative is not continuous.

40. Evaluate lim (x->e) ln(x).

A) 0 B) 1 C) e D) Undefined

41. What is the limit of f(x) = c (a constant) as x approaches infinity?

A) Infinity B) 0 C) c D) Undefined

42. If lim (x->a) f(x) = L, what is lim (x->a) [f(x)]^n?

A) L^n B) n*L C) L+n D) Undefined

43. A function f(x) is continuous on an open interval (a, b) if it is continuous at every point in the interval. Which of the following functions is continuous on (-infinity, infinity)?

A) f(x) = 1/x B) f(x) = tan(x) C) f(x) = x^2 + 3x - 5 D) f(x) = |x|/x

44. What is the value of lim (x->0) (sin(ax)/bx) where a and b are non-zero constants?

A) a/b B) b/a C) 1 D) 0

45. If lim (x->c) f(x) = L and L is a finite number, but f(c) is not defined, what type of discontinuity does f(x) have at x=c?

A) Jump discontinuity B) Infinite discontinuity C) Removable discontinuity D) Oscillating discontinuity

46. Evaluate lim (x->infinity) (3x^2 + 2x - 1)/(x^2 + 5).

A) 0 B) 3/5 C) 3 D) Infinity

47. What is the limit of the sequence a_n = (1 + 1/n)^n as n approaches infinity?

A) 0 B) 1 C) e D) Infinity

48. For lim (x->a) f(x) to exist, the left-hand limit and right-hand limit must be equal. What if f(x) = x for x<0 and f(x) = x+1 for x>=0? What is lim (x->0) f(x)?

A) 0 B) 1 C) Undefined D) -1

49. If a function f(x) is continuous everywhere, it is also called:

A) Differentiable everywhere B) Uniformly continuous C) An entire function D) A continuous function

50. What is lim (x->pi/2) cos(x)?

A) 0 B) 1 C) -1 D) pi/2

Limits and continuity - Online Test

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