Interact with the graph in order to determine a δ so that if |x - a| Let f(x) = 3x-1 and let x → 2, and suppose that ε = .5 Determine that δ so that |x - a| δ , then |f(x) - L|

Question

Visit this desmos link: https://www.desmos.com/calculator/ba7uaf4tey

Interact with the graph in order to determine a δ so that if |x - a| < δ , then |f(x) - L| < ε . Please make sure that your response is accurate to 3 decimal places.

Let f(x) = 3x-1 and let x → 2, and suppose that ε = .5

Determine that δ so that |x - a| < δ , then |f(x) - L| < ε

Answer 1

I believe you are trying to determine continuity for f(x) = 3x-1 at x = 2.

Given ε = 0.5, then we have:

|x- 2| < δ → |(3x - 2) - 5| < ε

|(3x-2)- 5| = |3x - 7| < ε = 0.5

Multiply both sides by 2 to get:

|6x - 14| < 1

|6(x-2) - 2| < 1

|6(x-2) - 2| = |6(x-2)| - 2 < 1

|(x-2)| < 3/6 = 0.5

We have then that δ = 0.5 applies.

However, constant functions are always continuous, and f(x) = x is also continuous at all points (both are easily shown), so f(x) = 3x-2 is continuous at all points.