Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.
lim x→1 (8x+5)=13

Verified step by step guidance

1. Step 1: Recall the precise definition of a limit. The limit of a function f(x) as x approaches a value c is L if for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε.
2. Step 2: Identify the function f(x) and the values of c and L from the given limit. Here, f(x) = 8x + 5, c = 1, and L = 13.
3. Step 3: Set up the inequality |f(x) - L| < ε using the function and limit value. This becomes |(8x + 5) - 13| < ε, which simplifies to |8x - 8| < ε.
4. Step 4: Simplify the inequality |8x - 8| < ε to find a relationship between x and ε. This simplifies to 8|x - 1| < ε, which further simplifies to |x - 1| < ε/8.
5. Step 5: Establish the relationship between ε and δ. From the inequality |x - 1| < ε/8, we can choose δ = ε/8. Therefore, for every ε > 0, if 0 < |x - 1| < δ, then |f(x) - 13| < ε, proving the limit.