Question

Find all the values of a, b ∈ R such that the function f(x) = sin(3x) if x > 0, ax if x = 0, arctan(1/x) if x < 0 is continuous?

Answer 1

To make the function continuous, we choose any value of a for the second part of the function and ensure that the arctan function approaches -pi/2 as x approaches 0 from the left.

Step-by-step explanation:

To find the values of a and b that make the function f(x) = sin(3x) if x > 0, ax if x = 0, arctan(1/x) if x < 0 continuous, we need to consider the three parts of the function separately.

1. For the function f(x) = sin(3x) if x > 0, we know that the sine function is continuous on its entire domain, so this part of the function is continuous for all values of x > 0.
2. For the function f(x) = ax if x = 0, we can choose any value of a to make this part continuous, as the graph will be a single point at x = 0.
3. For the function f(x) = arctan(1/x) if x < 0, we need to make sure that the limit of the function as x approaches 0 from the left is equal to the value of the function at x = 0. Since the arctan function approaches -pi/2 as x approaches 0 from the left, we need to choose a value of a that makes arctan(1/0) = -pi/2.

In conclusion, there are infinitely many values of a and b that make the function f(x) continuous, as we can choose any value for a in the second part of the function. The values of a and b are not restricted.