Question

Consider the function f(x)=x3+xx

Is f(x)
continuous at x=0?

Answer 1

Yes, f(x) is continuous at x = 0.

Explanation:

To determine if the function f(x) = x^3 + xx is continuous at x = 0, we need to check three conditions:

1. The function must be defined at x = 0.
2. The limit of the function as x approaches 0 from both the left and the right must exist.
3. The value of the function at x = 0 must be equal to the limit.

Let's evaluate these conditions for the given function:

Defined at x = 0:

The function is defined at x = 0 since f(0) = 0^3 + 0 = 0.

Limits as x approaches 0:

lim x→0− f(x) = lim x→0− (x^3 + xx) = lim x→0− x(x^2 + 1) = 0

lim x→0+ f(x) = lim x→0+ (x^3 + xx) = lim x→0+ x(x^2 + 1) = 0

Value of the function at x = 0:

f(0) = 0^3 + 0 = 0.

Since the function is defined at x = 0, the limits from the left and the right are equal, and the value of the function at x = 0 is equal to the limit, we can conclude that the function f(x) = x^3 + xx is continuous at x = 0.