Using the Formal Definition

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Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→−3 (x² − 9) / (x + 3) = −6

Accepted Answer

Welcome back, everyone. In this problem, we want to prove the limit statement that the limit of X2 minus 4 divided by X minus 2 as X approaches 2 equals 4. Which of the following choices for Delta in terms of Epsilon correctly completes the proof? A says it's that delta equals the root of Epsilon, B that it equals half of Epsilon, C2 Epsilon, and D Epsilon. Now how can we prove the limit statement and then figure out the value for delta? Well, recall what we know about the definition of the limit. We know that it basically says for every value of epsilon that's greater than 0, then there exists. A valley of delta. Greater than 0, such that, OK, such that. If The absolute value of X minus C is greater than 0 but less than Delta, then the absolute value of FF X minus L is going to be less than epsilon. Now if we look at our problem here, we know that we are given the value of FF X. We know that our function is X2 minus 4 divided by X minus 2. We also know that the value X approaches C equals 2, and we know that our limit L equals 4. So by applying the definition of the limit, then we are saying that for every epsilon. That's greater than 0, OK, then there exists. There exists. A value of. Delta, that's also greater than 0, OK, such that such that first. 0 is less than the absolute value of X minus 2, which is less than Delta, OK. And we also know that the absolute value of FFX X squad minus 4 divided by X minus 2, OK, is, sorry, minus, let me put my properly here. So FFX minus our function L, which minus our limit L, which is 4, is going to be less than epsilon. So now, how can we relate the information that we have in Epsilon to Delta in order to choose a value for delta that would correctly complete or proof? Well, let's try to do a bit of work with our function F of X here, OK? Now, for X not being equal to 2, if we factor the numerator, then we could rewrite X2 minus 4 divided by X minus 2. As X minus 2 multiplied by X + 2 divided by X minus 2 using the difference of 2 squares, which we can cancel X minus 2 to get X + 2. In other words, we could simplify our function to be X + 2. And now if we use that, OK, if we use FF X to be X + 2, then we're going to get the absolute value of X + 2 minus 4, which is going to be equal to the absolute value of X minus 2 because 2 minus 4 is -2 to be less than Epsilon. So no, in this case we must show that for every epsilon greater than 0, there exists a delta greater than 0, such that if the absolute value of X minus 2 is positive but less than Delta, then the absolute value of X minus 2 will be less than epsilon. So what value of delta do we need? Well, when we look at both of our inequalities here, OK. If the absolute value of X minus 2 is less than Delta and less than Epsilon, then we can choose delta or we can set delta to be equal to Epsilon, OK? Because if we set delta. To be equal to epsilon, then it's clear that whenever, whenever the absolute value of X minus 2 is less than delta but positive, then the absolute value of F of X minus 4 will be less than Epsilon. Therefore, delta should be equal to Epsilon. 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.

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→−3 (x² − 9) / (x + 3) = −6

Key Concepts

Limit Definition

Factoring and Simplifying Expressions

Substitution in Limits

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