Determine if the function f(x) = { x^2, x < 1; x + 1, x >= 1 } is continuous at x = 1.
Practice Questions
1 question
Q1
Determine if the function f(x) = { x^2, x < 1; x + 1, x >= 1 } is continuous at x = 1.
Yes
No
Depends on x
None of the above
Both sides equal 2 at x = 1, hence it is continuous.
Questions & Step-by-step Solutions
1 item
Q
Q: Determine if the function f(x) = { x^2, x < 1; x + 1, x >= 1 } is continuous at x = 1.
Solution: Both sides equal 2 at x = 1, hence it is continuous.
Steps: 5
Step 1: Identify the function f(x) which is defined in two parts: f(x) = x^2 when x < 1 and f(x) = x + 1 when x >= 1.
Step 2: Find the value of f(1) using the second part of the function since 1 is included in that part. Calculate f(1) = 1 + 1 = 2.
Step 3: Calculate the limit of f(x) as x approaches 1 from the left (x < 1). Use the first part of the function: limit as x approaches 1 from the left of f(x) = x^2 = 1^2 = 1.
Step 4: Calculate the limit of f(x) as x approaches 1 from the right (x >= 1). Use the second part of the function: limit as x approaches 1 from the right of f(x) = x + 1 = 1 + 1 = 2.
Step 5: Compare the left-hand limit (1) and the right-hand limit (2). Since they are not equal, the function is not continuous at x = 1.
