Using the Formal Definition

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Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→3 (3x − 7) = 2

Accepted Answer

Welcome back, everyone. In this problem, we want to prove the limit statement that the limit of 4 X + 3 as X approaches -2 equals -5 by choosing the correct value of delta in terms of epsilon that proves this limit. A says delta equals epsilon, B says it's 14th of Epsilon, C for Epsilon, and D Epsilon plus 1. Now how can we prove this limit statement? Well, let's set up the condition for Epsilon, OK? We know that for the limit to hold. For the limit to hold, then for every epsilon, that's positive. There exists a value of delta that's positive. Such that, OK. If The absolute value of X minus C is greater than 0 but less than delta. Then the difference, the absolute value of the difference between F of X and L, must be less than Epsilon. In our problem, we know we can, we can apply the idea that FFX or rather by reading our problem, we know that FFX. Is equal to 4 X + 24 X + 3, OK? Our limit L equals -5 and C or C equals -2. So by applying this definition, then this would imply. That 0 is less than the absolute value of X minus -2, which is. Less than Delta, OK. Which means Alright, let me write that out here, which means that the absolute value of X + 2 will be less than Delta. Likewise, Using our idea that the absolute value of FF X minus L is going to be less than Epsilon, then we can express this. As the absolute value of 4 X + 3, OK, minus 5. Which would be equal to 4 X plus 3, sorry, minus -5, my apologies, that L equals -5. And that is gonna be equal to 4X + 3 + 5, the absolute value of that, which is the absolute value of 4 X + 8, OK, which if we factor all 4 here, we could rewrite this as 4 multiplied by the absolute value of X plus 2. We know that this expression is going to be less than Epsilon. Now, let's look at both of our expressions here, including Delta and Epsilon. Can we equate them in some way? Can we set up the condition for them to hold? Well, OK, notice that. Notice that here from our expression for epsilon. If we're multiplied by the absolute value of X + 2 is less than Epsilon, then the absolute value of X + 2 will be less than 1/4 of Epsilon, OK? And no, that absolute value of X + 2 is also less than Delta. Thus, we can choose, OK, we can choose delta, sorry, to be equal to 4% of Epsilon to guarantee. OK. That whenever. The absolute value of X2 is positive but less than Delta. Then F of X minus l, which is 4 multiplied by the absolute value of X minus 2, is going to be less than epsilon. Thus, delta is going to be equal to 14th of epsilon. If we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→3 (3x − 7) = 2

Key Concepts

Limit Definition

Epsilon-Delta Proof

Function Behavior Near a Point

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