Understanding Limits and Function Behavior

The video tutorial explores the function f(x) = |x-3|/(x-3) and its limit as x approaches 3. Initially, the function is undefined at x=3, leading to a zero over zero form. The instructor rewrites the function for cases where x is greater than or less than 3, showing that f(x) equals 1 for x>3 and -1 for x<3. A graph is drawn to visualize these cases. The tutorial then analyzes the limits from both directions, concluding that the limit does not exist because the left-hand and right-hand limits are different.

What is the initial problem with the function f(x) as x approaches 3?

• The function is continuous at x = 3.

• The function is undefined at x = 3.

• The function is infinite at x = 3.

• The function is zero at x = 3.

How is the function f(x) rewritten for x greater than 3?

• f(x) = 1

• f(x) = 0

• f(x) = -1

• f(x) = x

What value does f(x) take for x less than 3?

• f(x) = x

• f(x) = 0

• f(x) = -1

• f(x) = 1

What does the graph of f(x) look like for x greater than 3?

• A vertical line at x = 3

• A diagonal line through the origin

• A horizontal line at y = -1

• A horizontal line at y = 1

What does the graph of f(x) look like for x less than 3?

• A diagonal line through the origin

• A horizontal line at y = 1

• A horizontal line at y = -1

• A vertical line at x = 3

What is the limit of f(x) as x approaches 3 from the left?

• 0

• Does not exist

• 1

• -1

What is the limit of f(x) as x approaches 3 from the right?

• 1

• -1

• 0

• Does not exist

Why does the limit of f(x) as x approaches 3 not exist?

• The function is zero at x = 3.

• The function is continuous at x = 3.

• The left-hand and right-hand limits are not equal.

• The function is undefined at x = 3.

What condition must be met for a limit to exist at a point?

• The function must be zero at that point.

• The function must be continuous at that point.

• The left-hand and right-hand limits must be equal.

• The function must be defined at that point.

What is the value of f(x) for x = 3?

• 1

• -1

• 0

• Undefined