For the function f(x)= {x^2, if x<1
{2x+1, if x> than or equal to 1.
Is f(x) continuous at x=1. Explain.
To determine if a function is continuous at a particular point, we need to confirm that three conditions are met:
1. The function exists at that point.
2. The limit of the function as x approaches that point exists.
3. The value of the function at that point is equal to the limit.
Let's evaluate these conditions for the given function f(x) = {x^2 if x < 1, 2x+1 if x ≥ 1} and determine if it is continuous at x = 1.
1. The function exists at x = 1 because both parts of the function are defined for this value. When x = 1, we have f(1) = 2(1) + 1 = 3.
2. Now, let's calculate the limit of the function as x approaches 1 from the left (x → 1-) and as x approaches 1 from the right (x → 1+).
As x approaches 1 from the left, f(x) = x^2. Therefore, the limit of f(x) as x approaches 1 from the left is lim(x → 1-) x^2 = 1^2 = 1.
As x approaches 1 from the right, f(x) = 2x + 1. Therefore, the limit of f(x) as x approaches 1 from the right is lim(x → 1+) (2x + 1) = 2(1) + 1 = 3.
3. Finally, we compare the value of the function at x = 1 with the limits we calculated in the previous step. We have f(1) = 3, which is equal to the limit from the right.
Since all three conditions are satisfied, the function f(x) is continuous at x = 1.