Limits and Continuity: Master Calculus Concepts with Interactive Quizzes

Question 1

Find the limit of (sin x) / x as x approaches 0.

Accepted Answer

1

Answer

Undefined

Answer

x

Answer

0

Question 2

Which of the following is the precise definition of the limit of a function f(x) as x approaches a?

Accepted Answer

For any ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε

Answer

f(a) = L

Answer

lim(x->a) f(x) = L means that f(x) is close to L when x is close to a

Question 3

If f(x) is continuous at x = a and g(x) is continuous at f(a), then the composite function h(x) = g(f(x)) is:

Accepted Answer

Continuous at x = a

Answer

Continuous only if f'(a) = g'(f(a))

Answer

Continuous only if g(f(a)) = 0

Answer

Discontinuous at x = a

Question 4

Which of the following is a necessary condition for a function to be differentiable at a point?

Accepted Answer

The function must be continuous at that point.

Answer

The function must be monotonic at that point.

Answer

The function must have a derivative at that point.

Question 5

Which of the following functions has an infinite discontinuity at x = 0?

Accepted Answer

f(x) = 1 / x

Answer

f(x) = sin(x) / x

Answer

f(x) = x^2 - 1

Answer

f(x) = |x|

Question 6

Determine whether the function f(x) = e^x / (x^2 + 1) is bounded on the interval [-1, 2].

Accepted Answer

Bounded

Answer

Unbounded

Answer

Neither bounded nor unbounded

Question 7

Find the points of discontinuity of the function f(x) = { 1 / x if x < 0, x^2 + 1 if x ≥ 0 }

Accepted Answer

x = 0

Answer

The function is continuous everywhere

Answer

x = 1

Answer

x = -1

Question 8

Determine the continuity of the function h(x) = 1 / (x^2 - 4) on the interval (-2, 2). Identify the points of discontinuity and explain why.

Accepted Answer

Discontinuous at x = -2 and x = 2

Answer

Discontinuous only at x = 2

Answer

Continuous on the interval

Answer

Discontinuous only at x = -2

Question 9

Determine the continuity of the function f(x) = 1 / sqrt(x - 3) on the interval (0, 3). Identify the point of discontinuity and explain why.

Accepted Answer

Discontinuous at x = 3

Answer

Discontinuous at x = 0

Answer

Continuous from the right

Answer

Continuous on the interval

Question 10

Find the limit of the function g(x) = (e^x - 1) / x as x approaches 0. Use L'Hopital's rule to evaluate the limit.

Accepted Answer

1

Answer

Infinity

Answer

Does not exist

Answer

0

Question 11

Determine the continuity of the function h(x) = { x^2 - 1 if x ≤ 0, x + 1 if x > 0 } at x = 0. Explain your reasoning.

Accepted Answer

Discontinuous

Answer

Continuous from the left

Answer

Continuous

Question 12

Determine the continuity of f(x) = x^2 - 1 at x = 1.

Accepted Answer

Continuous

Answer

Discontinuous jump

Answer

Discontinuous removable

Answer

Discontinuous infinite

Question 13

Find the horizontal asymptote: lim_(x->∞) (2x^2 + 3x - 5) / (x^2 - 1)

Accepted Answer

y = 2

Answer

y = 3

Answer

y = 1

Question 14

Determine the continuity of the function f(x) = (x^2 - 9) / (x - 3) at x = 3. Explain why the function is discontinuous at that point.

Accepted Answer

Discontinuous

Answer

Continuous

Answer

Discontinuous from the right

Answer

Discontinuous from the left

Question 15

Determine the continuity of the piecewise-defined function f(x) = { x^2 if x ≤ 2, 4 if x > 2 }. Explain your reasoning.

Accepted Answer

Discontinuous at x = 2

Answer

Continuous everywhere

Answer

Continuous at x = 2

Answer

Discontinuous at x = 0

Question 16

Find the one-sided limit of (x - 2) / (x^2 - 4) as x approaches 2 from the left. Show your work.

Accepted Answer

-1/2

Answer

1/2

Answer

0

Answer

Infinity

Question 17

Find the value of the constant c for which the function f(x) ={ x^2 + c if x < 2, 3x - 2 if x >= 2} is continuous at x = 2.

Accepted Answer

-1

Answer

1

Answer

2

Answer

0

Question 18

Which of the following functions has a vertical asymptote at x = 1?

Accepted Answer

f(x) = 1/(x - 1)

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 19

Is the function g(x) = |x - 2| continuous at x = 2?

Accepted Answer

Yes, the function is continuous at x = 2.

Answer

The function is differentiable but not continuous at x = 2.

Answer

The function is continuous but not differentiable at x = 2.

Answer

No, the function is discontinuous at x = 2.

Question 20

Is the function h(x) = |x| / x continuous at x = 0?

Accepted Answer

No, the function is discontinuous at x = 0.

Answer

The function is differentiable but not continuous at x = 0.

Answer

Yes, the function is continuous at x = 0.

Answer

The function is continuous but not differentiable at x = 0.

Question 21

Is the function g(x) = x³ - 2x² + x - 2 continuous for all real x?

Accepted Answer

Yes, the function is continuous for all real x.

Answer

The function is discontinuous at x = 1.

Answer

The function is discontinuous at x = 0.

Answer

The function is discontinuous at x = 2.