Use the definitions given in Exercise 57 to prove the followin...

Question

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^+ 1 /1 − x=−∞

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the limit as X approaches 5 from the right of 2 divided by 5 minus X. Use the definition of infinite limits to find the correct answer. A says it's infinity, B negative infinity, C says it's 0, and the D says it's 1. Now how are we gonna find this limit? Well, let's use what our problem told us here. Let's try to think, what do we know about the definition of infinite limits? Well, recall, OK, that if we have a limit, OK. As X approaches a value A from the right of FF X. It's equal to negative infinity if for every negative number in. OK. For every negative number and. There exists a value, a small value delta that's positive, such that, OK, such that F of X is less than N, OK. Whenever X minus A, the difference between X and the value or limit. is greater than 0, but less than Delta, OK? As a matter of fact, let me see if I can uh squeeze this. Oops, I can't. Let me just leave it here. So, this condition is true if we can satisfy this, or if we can find this value of delta for F of X. Now, what does that mean for us? Well, if we think about the limit we're concerned with, OK, this would mean then that for the limit. OK. As X approaches 5 from the right of 2 divided by 5 minus X, OK. If we can show that for any negative number, N. OK, in other words, we're applying our definition here. So if we can show that if for every for any negative number in there exists a value of delta. Greater than 0. Such that 2 divided by 5 minus X. F of X, OK, or function is less than N, OK. Whenever X minus 5 is greater than 0 but less than delta, then we know that our limit will be equal to negative infinity. So we need to prove, we need to show this in our solution. How are we going to do that? Well, first, let's try to rewrite our inequality. We know that our inequality is 2 divided by 5 minus X is less than N. Since X is approaching 5 from the right, OK, then that means X must be greater than 5. So that means 5 minus X is going to be less than 0 because if 5 is subtracting a value greater than 5, the result will be negative. So that means 2 divided by 5 minus X is negative. Since N is also negative, that means we can also rewrite our inequality then, OK, as. 2 divided by N is less than 5 minus X. Which means that we can say X is going to be less than 5 minus 2 divided by N. So now we've rearranged in equality to sell for X, but now remember we said X is greater than 5. So since X is greater than 5, that means X minus 5 is gonna be greater than 0 but less than 2 divided by the absolute value of N, OK, where that absolute value of N is going to be equal to negative N because earlier we said that N was negative. No, let delta, OK. Let's choose a value of delta. Let's let delta equal 2 divided by the absolute value of N, OK? Then that means. For every, for x minus 5 greater than 0, OK, but less than delta, then like we said earlier, X minus 5 will be greater than 0, but less than 2 divided by the absolute value of N. And now in this case this ensures, OK, this ensures. That 2 divided by 5 minus X is going to be less than N like we had started with, like we had said earlier. So now that we've shown that this condition will hold. Then if we define delta to be 2 divided by the absolute value of N, then by the definition of infinite limits for any negative number n. We can choose, yeah. Delta to be equal to 2 divided by the absolute value of N. Such that OK. 2 divided by 5 minus X is less than N. Whenever x minus 5 is greater than 0 but less than delta, and in that case, thus this means that the limit, since it satisfies the definition and the limit as X approaches5 from the right of 2 divided by 5 minus X will be equal to negative infinity. Therefore, when we look back on our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^+ 1 /1 − x=−∞

Key Concepts

Infinite Limits

One-Sided Limits

Understanding the Function's Behavior

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