Limits and Continuity in Functions

The video tutorial explains how to find the value of b such that the limit of f(x) as x approaches 2 exists. It covers the conditions for the existence of a limit, which requires that both one-sided limits exist and are equal. The tutorial demonstrates calculating the left-side limit and the right-side limit separately, then equates them to solve for b, resulting in b = 3.

What is the main objective of the problem discussed in the video?

• To find the maximum value of the function.

• To calculate the derivative of the function at x = 2.

• To determine the value of b for which the limit of f(x) as x approaches 2 exists.

• To find the value of x for which the function is continuous.

What must be true for the limit of a function to exist at a point?

• The function must have a maximum at that point.

• The one-sided limits must exist and be equal.

• The function must be differentiable at that point.

• The function must be continuous everywhere.

When approaching 2 from the left, which condition is used for f(x)?

• f(x) = 1/2 x + b

• f(x) = x^2 + b

• f(x) = -x + 6

• f(x) = x + b

What is the result of the limit as x approaches 2 from the left?

• b + 2

• b - 2

• b - 1

• b + 1

When approaching 2 from the right, which condition is used for f(x)?

• f(x) = x + b

• f(x) = -x + 6

• f(x) = 1/2 x + b

• f(x) = x^2 + b

What is the result of the limit as x approaches 2 from the right?

• 2

• 1

• 3

• 4

What equation is set up to find the value of b?

• b = 3

• b = 4

• 1 + b = 4

• 1 + b = 3

What is the value of b that makes the limit exist?

• 2

• 3

• 4

• 5