Understanding Limits and Continuity

The video tutorial explains the function g(x), defined piecewise with logarithmic expressions, and demonstrates how to find the limit of g(x) as x approaches 3. The process involves calculating the limit from both the left and right sides, ensuring continuity and consistency. The final result shows that the limit from both sides is the same, confirming the limit of g(x) at x=3.

What is the definition of g(x) when x is between 0 and 3?

• g(x) = log(3x)

• g(x) = 4 - x

• g(x) = x^2

• g(x) = log(9)

What is the definition of g(x) when x is greater than or equal to 3?

• g(x) = log(9)

• g(x) = log(3x)

• g(x) = 4 - x times log(9)

• g(x) = x^2

Why is it necessary to find both left-hand and right-hand limits?

• To find the derivative of the function

• To check if the function is continuous

• To ensure the overall limit exists and is the same from both sides

• To determine if the function is increasing

What is the left-hand limit of g(x) as x approaches 3?

• 9

• 3

• log(9)

• log(3)

What is the base of the logarithm used in the function g(x)?

• Base 5

• Base 10

• Base e

• Base 2

What is the right-hand limit of g(x) as x approaches 3?

• 3

• log(9)

• log(3)

• 9

What type of expression is 4 - x times log(9) for x greater than or equal to 3?

• A quadratic expression

• A logarithmic expression

• An exponential expression

• A linear expression

What does it mean if the left-hand and right-hand limits are equal?

• The function is not defined

• The function is discontinuous

• The overall limit exists and is equal to that value

• The function is decreasing

What is the value of log(9) approximately?

• Close to 3

• Close to 2

• Close to 1

• Close to 0

What is the overall limit of g(x) as x approaches 3?

• 3

• 9

• log(3)

• log(9)