Question

Let f (x )= 5x−1.

Find the largest δ so that |f(x)−f(a)∣<ϵ when ∣x−a∣<δ.
a) 5/ϵ
b) 6/ϵ
c) 7/ϵ
d) 8/ϵ

Answer 1

Option A). To find the largest δ (delta) so that |f(x)-f(a)| < ε when |x-a| < δ, divide ε by 5.

Step-by-step explanation:

To find the largest δ (delta) so that |f(x)-f(a)| < ε when |x-a| < δ, we need to analyze the given function and inequality. The function is f(x) = 5x-1 and the given inequality is |f(x)-f(a)| < ε. Let's break down each step:

1. Substitute f(x) into the inequality: |5x-1 - f(a)| < ε.
2. Expand the absolute value: |5x-1 - 5a+1| < ε.
3. Combine like terms: |5x-5a| < ε.
4. Use the triangle inequality to further simplify: |5(x-a)| < ε.
5. Divide both sides by 5: |x-a| < ε/5.

Therefore, the largest δ (delta) is ε/5. In this case, the correct answer is option a) 5/ε.