Use the precise definition of a limit to prove the following l...

Question

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→−3 |2x|=6 (Hint: Use the inequality ∥a|−|b∥≤|a−b|, which holds for all constants a and b (see Exercise 74).)

Accepted Answer

Welcome back, everyone. In this problem, we want to select the correct relationship between Epsilon and Delta to prove that the limit of the absolute value of 5 X as X approaches -8 equals 40 using the Epsilon minus delta definition of a limit. A says Delta E equals 5 divided by Epsilon, B it's Epsilon + 5. C, it's epsilon, and D says it's Epsilon divided by 5. No, we're trying to understand the relationship between Epsilon and Delta, but we'll need to use the Epsilon minus delta definition of a limit. What does this definition tell us? Well, basically, it says that given the limit, OK, as X approaches C of F of X equals L, where C is some value and L represents our limit, then if epsilon is greater than 0, no matter how small, basically very small values within the region of F of X. Then there exist small values of delta, OK. Which are also greater than 0 values within the region of C, OK, such that. Such that 0 is less than the difference between X minus C, which is less than delta. And if that is so, it implies that the absolute value of F of X minus L, the difference between the function and the limit, are also going to be less than Epsilon. So if we can prove, OK, this definition here, if we can use this definition to somehow relate Epsilon to delta, then we should be able to show, or we will be sure rather that the limit exists, OK? So let's go ahead and see if we can do that. How can we show Or how can we express Epsilon in terms of delta using this relationship? Well, let's let C be equal to -8 based on our limit here for our function. And let's also letl the limit be equal to 40. Then for The values of delta, OK, is going to be between 0 and the absolute value of X minus -8, which is going to be X + 8, which is gonna be less than Delta. And no for epsilon greater than 0, that is a positive values of epsilon, no matter how small, is going to be the absolute value of our function, the absolute value of 5 X minus our limit 40, and that absolute value is going to be less than epsilon. No, here, if we rewrite. OK, minus 40, or sorry, if we were at 40 as the absolute value of -40, OK, being less than epsilon, then we can apply the inequality that says if we're finding the absolute value of the difference between two absolute values, it's going to be less than or equal to the absolute value of the difference between those values inside the absolute value. In other words, a minus B, OK. So that tells us then that if we have 5 X here and minus -40. Then we could rewrite this as the absolute value of 5X minus -40 or 5X plus 40. I know this is going to be less than Epsilon. Now, here, if we factor 5 from our expression, OK. Then this would imply that we could rewrite this as 5 multiplied by the absolute value of X + 8 as less than Epsilon. And now, in that case, if we divide both sides by 5, the absolute value of X + 8 must then be less than epsilon divided by 5. But do we know anything else about the absolute value of X + 8? Yes, we do. We also know that the absolute value of X +8 must be less than Delta. So if it's less than Delta and less than Epsilon divided by 5, this would imply, OK, let's write this here. Since it's less than Delta, eh, then that must mean that Delta is going to be equal to Epsilon divided by 5. In other words, this has to be the relationship between Delta and Epsilon for the limit to exist. If we look back at our answer choices, that means D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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→−3 |2x|=6 (Hint: Use the inequality ∥a|−|b∥≤|a−b|, which holds for all constants a and b (see Exercise 74).)

Key Concepts

Limit Definition

Absolute Value and Inequalities

Epsilon-Delta Relationship

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