Prove the statement using the ε, δ definition of a limit. lim x → 1 4 + 2x 3 = 2 Given ε > 0, we need δ ---Select--- such that if 0 < |x − 3| < δ, then 2 + 1 3 x − 3 ---Select--- . But 2 + 1 3 x − 3 < ε ⇔ 1 3 x − 1 < ε ⇔ 1 3 |x − 3| < ε ⇔ |x − 3| < ---Select--- . So if we choose δ = ---Select--- then 0 < |x − 3| < δ ⇒ 2 + 1 3 x − 3 < ε. Thus, lim x → 3 2 + 1 3 x = 3 by the definition of a limit.

You want to prove

By the definition of the limit, this means that for any , we can find such that

.

In order to get the inequality we want, we need to have

so we can pick .

So here's the proof: Let be given, and let . Then

as required.

walter needs to make at least 5 more cookies but no more than 40. 5 ≤ x ≤ 40

step-by-step explanation:

20 ≤ x + 15 ≤ 55

subtract 15 from all parts

20-15 ≤ x + 15-15 ≤ 55-15

5 ≤ x ≤ 40

this means how many more cookies he has to make

walter needs to make at least 5 more cookies, but no more than 40 cookies; 5 ≤ x ≤ 40