Calculus Continuity Problems

Problem 4

• To determine continuity, we need to check if the function's value at x = 1 matches the limit of the function as x approaches 1.
• For x < 1, the function is defined by x² - 1.
• For x > 1, the function is defined by x² - x.
• The limit as x approaches 1 from the left is 1² - 1 = 0.
• The limit as x approaches 1 from the right is 1² - 1 = 0.
• Therefore, g(x) is continuous at x = 1 if g(1) = 0.
• However, g(1) is defined as 1² - 1 = 0.
• g(x) is continuous for all values of k because the function is continuous at x = 1 regardless of the value of k.
• There is no value of x at which g(x) is discontinuous.

Problem 3

• To ensure continuity, we need to make sure the function's value at x = 1 matches the limit of the function as x approaches 1.
• For x < 1, the function is defined by x² + 2.
• For x > 1, the function is defined by 3x - 1.
• The limit as x approaches 1 from the left is 1² + 2 = 3.
• The limit as x approaches 1 from the right is 3(1) - 1 = 2.
• Therefore, f(x) is continuous for all values of x if f(1) = 3.
• To achieve this, we need to set k = 3.
• Hence, f(x) is continuous for all x when k = 3.

This quiz focuses on determining the continuity of functions through limits and function values. You will work with piecewise functions and analyze their behavior around a critical point, specifically at x = 1. Test your understanding of the concepts of continuity and limit in calculus.