Use formal definitions to prove the limit statements in Exerci...

Question

Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → 0 (1 / |x|) = ∞

Accepted Answer

Welcome back, everyone. For this problem we want to prove the limit statement that the limit of 2 divided by X + 1 as X approaches -1 equals infinity by choosing the correct delta that proves the given limit. A says delta is 1/2 of M. B says it's 2 divided by M. C 3 divided by M. and the D 1 divided by 2 M. Now how can we prove this limit statement? What do we know? Well, recall that by definition the definition basically tells us that. If, OK, for every value of M, a large number greater than 0, OK, there exists. Small value delta, that's also greater than 0, such that, OK. Whenever, whenever 0 is less than the absolute value of X minus C, which is less than Delta, OK. In other words, the difference between X and C, the absolute value is positive but less than Delta, then. We're going to have our function F of X, OK, that's greater than M. Now, in this case, for our problem, OK, we know that FF X is equal to 2 divided by X + 1, OK. See, OK. C is equal to -1, OK. So by applying the definition, then we're saying that for every, for every M that's greater than 0, we want to show that there exists. A delta greater than 0, such that, OK. We know we're going by the definition. Whenever the 0 is less than the absolute value of X minus -1, which is equal to the absolute value of X + 1 is less than delta. Then FF X, which is 2 divided by X + 1, must be greater than M. So this is ultimately what we want to show, but to show that we'll need to express, we'll need to express delta in terms of M. We'll need to relate to both of them so that we'll be able to solve, OK. So now, let's say that this is going to be 2 divided by the absolute value of X + 1 greater than M. How can we relate that to our to what we know about Delta? Well, let's start with this part of our of our expression of our definition, OK? Given 2 divided by the absolute value of X + 1 is greater than M. Then if we rearrange this inequality, multiplying both sides by the absolute value of X + 1. Then we're going to get 2 to be greater than M multiplied by the absolute value of X + 1, which means then that if we divide through by M, the absolute value of X + 1 is going to be less than 2 divided by M. No remember earlier we said the absolute value of X + 1 must be less than delta. So if it's less than Delta and it's less than 2 divided by m to satisfy the definition, we can choose, OK, we can then choose delta to be equal to 2 divided by M. No, whenever the absolute value of X + 1 is between 0 and delta, OK, then. Then that means. That the absolute value of X + 1 is going to be between 0 and 2 divided by M. And thus this implies that the limit will tend to infinity. So now we have formally proved the given limit statement since we have found such a delta for every positive value of M. Thus, delta equals 2 divided by M. and 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.

Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → 0 (1 / |x|) = ∞

Key Concepts

Limit Definition

Behavior of Functions Near Zero

Infinity in Limits

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