Limits and continuity - Question Bank

1. If lim (x->a) f(x) = L, what is lim (x->a) [sqrt(f(x))]? (Assume f(x) >= 0 near a and L >= 0)
A) sqrt(L)
B) L^2
C) L/2
D) Undefined
2. Consider the function f(x) = (x^2 - 9)/(x - 3). Is f(x) continuous at x=3?
A) Yes, because the limit exists.
B) No, because f(3) is undefined.
C) Yes, if we define f(3) = 6.
D) No, because it is a rational function.
3. What is lim (x->pi/2) cos(x)?
A) 0
B) 1
C) -1
D) pi/2
4. 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
5. 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
6. 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
7. Evaluate lim (x->infinity) (3x^2 + 2x - 1)/(x^2 + 5).
A) 0
B) 3/5
C) 3
D) Infinity
8. 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
9. 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
10. 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
11. 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
12. What is the limit of f(x) = c (a constant) as x approaches infinity?
A) Infinity
B) 0
C) c
D) Undefined
13. Evaluate lim (x->e) ln(x).
A) 0
B) 1
C) e
D) Undefined
14. 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.
15. 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.
16. 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
17. What is lim (x->0) (1 - cos(x))/x?
A) 0
B) 1
C) 1/2
D) Undefined
18. What is the limit of the function f(x) = 2x + 3 as x approaches 5?
A) 2
B) 3
C) 5
D) 13
19. Evaluate lim (x->1) (x^3 - 1)/(x - 1).
A) 0
B) 1
C) 3
D) Infinity
20. If f(x) is continuous at x=c, then lim (x->c) f(x) = ?
A) f'(c)
B) f(c)
C) 0
D) Undefined
21. What is the value of lim (x->infinity) (sin(x)/x)?
A) 1
B) 0
C) Infinity
D) Undefined
22. 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.
23. 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
24. 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).
25. 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
26. What is the limit of a polynomial function P(x) as x approaches 'a'?
A) P'(a)
B) P(a)
C) 0
D) Infinity
27. 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
28. What is lim (x->0) (tan(x)/x)?
A) 0
B) 1
C) Infinity
D) Undefined
29. Consider the function f(x) = 1/x. What is lim (x->0) f(x)?
A) 0
B) 1
C) Infinity
D) Does not exist
30. What is lim (x->0) cos(x)?
A) 0
B) 1
C) Undefined
D) pi/2
31. Evaluate lim (x->pi) sin(x).
A) -1
B) 0
C) 1
D) pi
32. 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
33. 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
34. What is the right-hand limit of f(x) = x^2 at x=3?
A) 3
B) 6
C) 9
D) Undefined
35. What is the left-hand limit of f(x) = x^2 at x=3?
A) 3
B) 6
C) 9
D) Undefined
36. 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
37. What is the limit of x^n as x approaches infinity, where n > 0?
A) 0
B) 1
C) Infinity
D) Depends on n
38. Evaluate lim (x->infinity) (1/x).
A) Infinity
B) 1
C) 0
D) Undefined
39. What is the value of lim (x->0) (e^x - 1)/x?
A) 0
B) 1
C) e
D) Undefined
40. 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.
41. 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.
42. 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
43. 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
44. 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
45. The limit of a constant function f(x) = k as x approaches any value 'a' is:
A) x
B) a
C) k
D) 0
46. 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
47. Evaluate lim (x->2) (x^2 - 4)/(x - 2).
A) 0
B) 2
C) 4
D) Undefined
48. What is the value of lim (x->0) (sin(x)/x)?
A) 0
B) 1
C) Undefined
D) Infinity
49. 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'.
50. 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
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