The function f(x) = { x^2, x < 0; 1, x = 0; x + 1, x > 0 } is continuous at x = 0?
Practice Questions
1 question
Q1
The function f(x) = { x^2, x < 0; 1, x = 0; x + 1, x > 0 } is continuous at x = 0?
Yes
No
Only from the right
Only from the left
Limit as x approaches 0 from left is 0, and f(0) = 1, hence it is not continuous at x = 0.
Questions & Step-by-step Solutions
1 item
Q
Q: The function f(x) = { x^2, x < 0; 1, x = 0; x + 1, x > 0 } is continuous at x = 0?
Solution: Limit as x approaches 0 from left is 0, and f(0) = 1, hence it is not continuous at x = 0.
Steps: 5
Step 1: Identify the function f(x) which is defined in three parts: f(x) = x^2 when x < 0, f(x) = 1 when x = 0, and f(x) = x + 1 when x > 0.
Step 2: Find the limit of f(x) as x approaches 0 from the left (x < 0). This means we use the part of the function for x < 0, which is f(x) = x^2.
Step 3: Calculate the limit as x approaches 0 from the left: limit as x -> 0- of f(x) = limit as x -> 0- of x^2 = 0.
Step 4: Find the value of the function at x = 0. According to the function definition, f(0) = 1.
Step 5: Compare the limit from the left (which is 0) with the value of the function at x = 0 (which is 1). Since 0 is not equal to 1, the function is not continuous at x = 0.
