Use Theorem 3.10 to evaluate the following limits.
lim x...

Question

Use Theorem 3.10 to evaluate the following limits.

lim x🠂2 (sin (x-2)) / (x² - 4)

Accepted Answer

Welcome back, everyone. In this problem, we want to use the following theorem to evaluate the limit as X approaches 9 of the sin of X minus 9 divided by X minus 81. The theorem that the limit of sin x divided by X as X approaches 0 is equal to 1. A says the limit is 136, B 1/18th, C9, and D says it's 36. Now how is this theorem supposed to help us to evaluate our limit? Well, let's think about what this theorem is saying here. Basically, what it's saying is that the limit as X approaches 0 of the sign of a function divided by its argument is equal to 1. So if we can get the argument of our sine function in this case X minus 9 in our denominator, then we should be able to apply the limit and thus evaluate it. So let's go ahead and try to do that. Now, let's take a good look at our denominator, OK? Now, in our denominator. OK. Then notice that X2 minus 81 is the difference of 2 squares, which means it can be rewritten as X + 9 multiplied by X minus 9. In that case, then that means we can take our our limit, the limit as X approaches 9 of the sign of X minus 9. And write it as being divided by X minus 9 multiplied by X + 9. Now, we could take this a step further, OK. And rewrite this as our limit, OK, as X approaches 9. Of the sign of X minus 9. Divided by X minus 9 multiplied by 1 divided by X + 9. Now why would we write it this way? Well, no, by looking you may notice that we've satisfied our our our theorem or we've written it in the form of our theorem. In other words, we have the sign or sorry, the argument of the sign function. Is dividing this the sign function, OK? If that makes sense. Let me say that another way. The sign of the argument is being divided by the argument X minus 9. So in this case, if we apply our limit, OK. Then that means. No, we can rewrite this as the limit as x approaches 9 multiplied by the sign of x minus 9 divided by X minus 9 multiplied by the limit as X approaches 9 of 1 divided by X + 9, and no, we can rewrite our first term as 1. So that's 1 multiplied by the limit as X approaches 9 of 1 divided by X + 9. No, we can directly substitute 9 into a limit, which is gonna be 1 divided by 9 + 9, which equals 1/18th. So by using the theorem, the limit is going to be 1/18th. 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 Theorem 3.10 to evaluate the following limits.
lim x🠂2 (sin (x-2)) / (x² - 4)

Key Concepts

Limits

Theorem 3.10 (L'Hôpital's Rule)

Trigonometric Limits

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