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→7 f(x)=9, where f(x)={3x−12 if x≤7

x+2 if x>7

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with limits. This problem says select the correct relationship between epsilon and Delta to prove that the limit as X approaches 9 of F of X is equal to 88, where the function F of X is equal to the piece wise function. Which we have 9 X + 7 if X is less than or equal to 9, and X + 79 if X is greater than 9. And we need to prove this limit using the epsilon-delta definition of a limit. So we're given 4 possible choices as our answers. For choice A, we have Delta is equal to 9 plus epsilon. For choice B, we have Delta is equal to epsilon divided by 9. For choice C, we have Delta is equal to 9 divided by epsilon, and for choice D, we have Delta is equal to 9 multiplied by epsilon. So, we need to prove the limit as Xstein of F of X is equal to 88, using the epsilon-delta definition. So, for that, we need to 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 9, and the 9 here comes from the value that X is approaching in the limit. And that also has to be less than Delta. So if this statement is true, Then We also need the absolute value of our function, which will use the first part of our piece-wise function, which is the absolute value of 9 X + 7 minus 88, which is the value that our limit is approaching. And that quantity has to be less than epsilon. Now, what we can do is simplify the quantity that is inside the absolute value signs, or there are second inequality there with epsilon. And so the absolute value of 9 X. Plus 7 minus 88. is equal to the absolute value of 9 X. -81. And I can factor out a 9 out of both of those terms, and so this is going to be equal to 9, multiplied by the absolute value of X minus 9. So that means that our inequality is 9, multiplied by the absolute value of X minus 9. is less than epsilon. So, I can divide both sides by 9, and I'll get the absolute value of X minus 9. is less than Epsilon divided by 9. So if I compare this inequality. To the one that contains x and delta. For both of these statements to be true, we see that we need to choose delta to be equal to epsilon divided by 9. And so this is the relationship that we're looking for. And if we look at our answer choices, the correct answer choice corresponds to answer choice B. 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→7 f(x)=9, where f(x)={3x−12 if x≤7
x+2 if x>7

Key Concepts

Definition of a Limit

Piecewise Functions

Epsilon-Delta Proof

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