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→2 (x^2+3x)=10

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with limits. This problem says to select the correct relationship between epsilon and Delta to prove that the limit as X approaches 3 of the quantity of X2 + 3 X is equal to 18 using the epsilon-delta definition of a limit. We're given 4 possible choices as our answers. For choice A, we have Delta is equal to the minimum of 5 and epsilon divided by 7. For choice B, we have Delta is equal to the minimum of 5 and epsilon divided by 10. For choice C, we have Delta is equal to the maximum of 5 and epsilon divided by 10, and for choice D, we have Delta is equal to the minimum of 1 and epsilon divided by 10. Now, the question asks us to prove this limit, as X approaches 3 of the quantity of X2 + 3 X is equal to 18, using the epsilon-delta definition of a limit. So, to use the definition, we need to first let epsilon be greater than 0, and we need to find a delta greater than 0. Such that if 0 is less than the absolute value of X minus 3 is less than delta, where the 3 there comes from the value that X is approaching in the limit. And so if this statement is true, then it follows that the absolute value of our function, which in this case is X2 + 3 X. Minus the value that the limit is evaluating to, which in this case is 18, has to be less than epsilon. Now we can factor the quantity inside the absolute value sign. So, doing that, we'll have the absolute value of the quantity of X plus 6. In quantity, multiplied by the quantity of X minus 3 in quantity, and that has to be less than epsilon. And we can just take the absolute value of each of those factors, just to simplify the case of it. So we'll have the absolute value of X + 6. Multiplied by the absolute value of X minus 3, and that has to be less than epsilon. Now at this point we need to make an assumption. And we need to assume That delta is less than or equal to one. And if we make this assumption, that means that the absolute value of X minus 3 has to be less than 1. And this will put limits on a value of X, so X can be between 2 and 4. That means that 2 is less than X, which is less than 4. Now, since X can be between 2 and 4, that means our factor of X plus 6 can be between 8 and 10. That means that 8 has to be less than X + 6. Which is less than 10. And since X + 6 has to be less than 10, then the quantity of the absolute value of X + 6. Multiplied by the absolute value of X minus 3. Has to be less than 10, multiplied by the absolute value of X minus 3. So that means that 10. Multiplied by the absolute value of X minus 3. Has to be less than Absalon. And so we can divide both sides by 10. In doing so, we will get the absolute value of X minus 3 is less than epsilon divided by 10. And so now we have two options. We have the absolute value of X minus 3 has to be less than epsilon divided by 10, and also the absolute value of X minus 3 has to be less than 1. We also have the statement that the absolute value of X minus 3 has to be less than delta. So What we need to do is to choose Delta to be the smallest of 1 and epsilon divided by 10. So we're going to want Delta to be equal to the minimum. Of either 1 or Epsilon divided by 10. So that is the relationship between Delta and Epsilon that I'm looking for. And if I look at my answer choices, this corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

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→2 (x^2+3x)=10

Key Concepts

Limit Definition

Epsilon-Delta Relationship

Polynomial Functions

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