Final answer:
The prove the limits using the ε,δ definition, for the first limit, lim(x→1) (3 / (2+4x)) = 2, we can find a δ < 1/4. For the second limit, lim(x→10) (3 - (5/(4x))) = -5, we can find a δ < 1/40.
Step-by-step explanation:
In order to prove the limits using the ε,δ definition, we need to show that for any given ε (epsilon), we can find a δ (delta) such that if the distance between x and the limit point is less than δ, then the distance between f(x) and the limit is less than ε. Let's start with the first limit:
a) limx→1 (3 / (2+4x)) = 2
We need to find a δ such that if |x-1| < δ, then |f(x)-2| < ε. Let's set a restriction on δ: δ < 1/4. Now, we can proceed with the proof:
|f(x)-2| = |(3 / (2+4x)) - 2| = |3 - 2(2+4x)| / (2+4x) = |3 - 4 - 8x| / (2+4x) = |-1 - 8x| / (2+4x) = (8x + 1) / (2+4x) ≤ 9/6 = 3/2, if we set δ = 1/4.
This shows that for any ε > 0, we can find a δ > 0 such that if |x-1| < δ, then |(3 / (2+4x)) - 2| < ε.
b) limx→10 (3 - (5/(4x))) = -5
Using a similar approach, we can set a restriction on δ: δ < 1/40. The proof follows in a similar manner as in part a), and we can conclude that for any ε > 0, we can find a δ > 0 such that if |x-10| < δ, then |(3 - (5/(4x))) - (-5)| < ε.