Types of Discontinuities and Limits

The video tutorial explains how to prove a function's continuity at a point using a three-step test. It involves checking if the function is defined at the point, if the limit exists as x approaches the point from both sides, and if the limit equals the function's value at that point. Examples are provided to illustrate continuity at x=2 and discontinuities at x=3 and x=-1, highlighting jump and removable discontinuities.

What is the first step in the three-step continuity test?

• Prove that the limit exists.

• Show that the function is defined at a point.

• Check if the left and right limits are equal.

• Determine if the function is differentiable.

What must be true for the limit to exist at a point according to the three-step continuity test?

• The function must be decreasing.

• The function must be increasing.

• The left and right limits must be equal.

• The function must be differentiable.

What is the value of f(2) for the given function in the example?

• 1

• 2

• 4

• 3

Why is the function continuous at x = 2?

• The function is not defined at x = 2.

• The left and right limits are not equal.

• The limit as x approaches 2 is equal to f(2).

• The function is differentiable at x = 2.

What type of discontinuity is present at x = 3?

• Jump discontinuity

• No discontinuity

• Infinite discontinuity

• Removable discontinuity

What is the value of f(3) for the given function in the example?

• 13

• 11

• 9

• 7

What is the value of f(1) in the new example?

• 5

• 4

• 3

• 6

What type of discontinuity is identified at x = 1?

• Jump discontinuity

• Removable discontinuity

• Infinite discontinuity

• No discontinuity

Which type of discontinuity occurs when the limit exists but the function value is different?

• Oscillating discontinuity

• Infinite discontinuity

• Removable discontinuity

• Jump discontinuity

When does an infinite discontinuity occur?

• When the function is not defined at a point.

• When the left and right limits are equal.

• When the function is continuous.

• When the limits approach infinity.