Determine if the function f(x) = { x^2, x < 1; 3, x = 1; 2x, x > 1 } is continuous at x = 1.
Practice Questions
1 question
Q1
Determine if the function f(x) = { x^2, x < 1; 3, x = 1; 2x, x > 1 } is continuous at x = 1.
Continuous
Not continuous
Depends on k
None of the above
At x = 1, f(1) = 3, but lim x->1- f(x) = 1 and lim x->1+ f(x) = 2. Thus, it is not continuous.
Questions & Step-by-step Solutions
1 item
Q
Q: Determine if the function f(x) = { x^2, x < 1; 3, x = 1; 2x, x > 1 } is continuous at x = 1.
Solution: At x = 1, f(1) = 3, but lim x->1- f(x) = 1 and lim x->1+ f(x) = 2. Thus, it is not continuous.
Steps: 6
Step 1: Identify the function f(x) and the point of interest, which is x = 1.
Step 2: Find the value of the function at x = 1. This is f(1) = 3.
Step 3: Calculate the left-hand limit as x approaches 1 from the left (x < 1). This means using the part of the function for x < 1, which is f(x) = x^2. So, lim x->1- f(x) = 1^2 = 1.
Step 4: Calculate the right-hand limit as x approaches 1 from the right (x > 1). This means using the part of the function for x > 1, which is f(x) = 2x. So, lim x->1+ f(x) = 2*1 = 2.
Step 5: Compare the value of the function at x = 1 (which is 3) with the left-hand limit (1) and the right-hand limit (2).
Step 6: Since f(1) = 3 is not equal to the left-hand limit (1) and the right-hand limit (2), the function is not continuous at x = 1.
