40.4k views
4 votes
Consider the function f(x)=x3+xx

Is f(x)
continuous at x=0?

User Ali Gajani
by
7.7k points

1 Answer

4 votes

Answer:

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.

User Crakama
by
8.1k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories