Question

Prove each of the following functions in continuous at x₀ by verifying the ϵ-δ property of Theorem 17.2. (a) f(x)=x², x₀=2;

Answer 1

To prove that f(x) = x² is continuous at x₀ = 2, we need to verify the ϵ-δ property of Theorem 17.2. By applying this property, we can find a positive δ such that if |x - 2| < δ, then |(x² - 2²)| < ϵ. Therefore, the function f(x) = x² is continuous at x₀ = 2.

Step-by-step explanation:

To prove that f(x) = x² is continuous at x₀ = 2, we need to verify the ϵ-δ property of Theorem 17.2. The ϵ-δ property states that for every positive ϵ (epsilon), there exists a positive δ (delta) such that if |x - x₀| < δ, then |f(x) - f(x₀)| < ϵ.

Let's apply this property to the given function: f(x) = x² and x₀ = 2.

We need to find a δ such that if |x - 2| < δ, then |(x² - 2²)| < ϵ.

Let's choose a positive δ such that δ < 1. This means that if |x - 2| < δ, then |x - 2| < 1. Solving this inequality gives us -1 < (x - 2) < 1, which simplifies to 1 < x < 3.

Now, let's consider f(x) - f(x₀) = (x² - 4). If we substitute x = 2 + δ/2, we get (2 + δ/2)² - 4 = δ²/4. Since δ < 1, δ²/4 < ϵ.

Therefore, we have proved that for every positive ϵ, there exists a positive δ such that if |x - 2| < δ, then |(x² - 2²)| < ϵ. Hence, the function f(x) = x² is continuous at x₀ = 2.