Question

Use the ϵ-δ definition of limits to prove that limx→1 4x + 2 =
6

Answer 1

To prove the limit using the ϵ-δ definition, we need to show that for every positive ϵ, there exists a positive δ such that if |x - 1| < δ, then |(4x + 2) - 6| < ϵ.

Step-by-step explanation:

To prove that limx→1 (4x + 2) = 6 using the ϵ-δ definition of limits, we need to show that for every positive ϵ, there exists a positive δ such that if |x - 1| < δ, then |(4x + 2) - 6| < ϵ.

1. Let ϵ > 0 be given.
2. Choose δ = ϵ/4.
3. Now, suppose |x - 1| < δ.
4. Then, |(4x + 2) - 6| = |4x - 4| = 4|x - 1| < 4(δ) = 4(ϵ/4) = ϵ.

Therefore, by the definition of a limit, we have proven that limx→1 (4x + 2) = 6.