Understanding Limits and Discontinuities

The video tutorial explains a piecewise continuous function f(x) defined over two intervals. It explores the limit of f(x) as x approaches 2, which is the boundary between the intervals. The left-hand limit is evaluated using the natural log of x, while the right-hand limit uses x^2 times the natural log of x. Despite both limits existing, they differ, indicating a jump discontinuity at x=2, meaning the limit does not exist.

What is the function f(x) defined as for 0 < x ≤ 2?

• x^2 * ln(x)

• ln(x)

• e^x

• x^3

What is the function f(x) defined as for x > 2?

• x + ln(x)

• x^2 * ln(x)

• ln(x)

• x^3

What is the value of f(2) according to the piecewise function?

• 2

• 4

• ln(2)

• 2^2 * ln(2)

What is the left-hand limit of f(x) as x approaches 2?

• 4

• 2

• ln(2)

• 2^2 * ln(2)

Which interval is used to evaluate the left-hand limit as x approaches 2?

• x < 0

• x > 2

• 0 < x ≤ 2

• x = 2

What is the right-hand limit of f(x) as x approaches 2?

• ln(2)

• 4

• 2

• 2^2 * ln(2)

Which interval is used to evaluate the right-hand limit as x approaches 2?

• 0 < x ≤ 2

• x > 2

• x < 0

• x = 2

What type of discontinuity is present at x = 2?

• No discontinuity

• Infinite discontinuity

• Jump discontinuity

• Removable discontinuity

Why does the limit of f(x) as x approaches 2 not exist?

• The function is not piecewise

• The function is not defined at x = 2

• The function is continuous at x = 2

• The left-hand and right-hand limits are different

What would you observe in the graph of f(x) at x = 2?

• A smooth curve

• A vertical asymptote

• A jump in the graph

• A hole in the graph