Is the function f(x) = { x^3, x < 1; 2x + 1, x >= 1 } continuous at x = 1?
Practice Questions
1 question
Q1
Is the function f(x) = { x^3, x < 1; 2x + 1, x >= 1 } continuous at x = 1?
Yes
No
Only left continuous
Only right continuous
Both limits as x approaches 1 from the left and right are equal to 2, hence f(x) is continuous at x = 1.
Questions & Step-by-step Solutions
1 item
Q
Q: Is the function f(x) = { x^3, x < 1; 2x + 1, x >= 1 } continuous at x = 1?
Solution: Both limits as x approaches 1 from the left and right are equal to 2, hence f(x) is continuous at x = 1.
Steps: 4
Step 1: Identify the function f(x) which is defined in two parts: f(x) = x^3 when x is less than 1, and f(x) = 2x + 1 when x is greater than or equal to 1.
Step 2: Find the limit of f(x) as x approaches 1 from the left (x < 1). This means we use the first part of the function: f(x) = x^3. So, calculate the limit: lim (x -> 1-) f(x) = 1^3 = 1.
Step 3: Find the limit of f(x) as x approaches 1 from the right (x >= 1). This means we use the second part of the function: f(x) = 2x + 1. So, calculate the limit: lim (x -> 1+) f(x) = 2(1) + 1 = 3.
Step 4: Compare the two limits from Step 2 and Step 3. The left limit is 1 and the right limit is 3. Since they are not equal, the function is not continuous at x = 1.
