Determine the following limits.
lim x→−∞ (5 + 100/...

Question

Determine the following limits.

lim x→−∞ (5 + 100/x + sin⁴ x³ / x²)

Accepted Answer

Welcome back, everyone. In this problem, we want to find the limit as X approaches infinity of 12 + 35 divided by X plus the sign of X to the 5th, race to the 6th divided by X the 4th. A says it's infinity, B 13, C 12, and B says it's 47. Now, for this problem, OK, since we're adding all the terms in the expression for the limit, we can find the limit of each expression. In other words, this will be equal to the limit as X approaches infinity of 12. Plus the limit as x approaches infinity of 35 divided by X. Plus the limit as x approaches infinity of sine x to the 5th to the 6th. Divided by X to the 4th. Now, let's see if we can find these limits one by one. We know that the limit of our constant term here is going to be equal to the constant 12. Next, as x approaches infinity for 35 divided by X, then the denominator will become arbitrarily large and thus this limit will be equal to 0. Now the final question is what will be the limit as x approaches infinity of sine x to the 5th race to the 6th divided by X the 4th. Well, let's take this expression apart. Let's let's look at the nature of our assigned term here. Now we know, OK, so let me just put 3 dots. We're going to revisit this. We know, OK, that. Sign the sign of the sign of X to the 5th to the 6th is going to be between 0 and 1, OK? Now, if we divide all our terms by X to the 4th, it's going to be 0, divided by X to the 4th which stays 0. Is less than equal to sin x 53 to the 6th divided by X 4th, which is less than R equal to 1 divided by X to the 4th, OK? Now what do we know? What do we know here? Well, and we know that since this term is between two other terms, we can apply the squeeze theorem to find the limit. Recall that by the squeeze theorem. If FF X. Is less than or equal to geo, which is less than or equal to H X. In other words, if geoffX is squeezed between F of X and HF X, OK? When X is near A. Except possibly at A. OK, except possibly at A. And the limit, OK. As X approaches A of F of X is equal to the limit as X approaches A of H X, which is equal to some limit L, then that must mean that the limit. As x approaches A of G of X is also going to be equal to L. So if we can find the limits of our two terms on the end here, 0 and 1 divided by X the 4th, and if they are the same, then that's also going to be the limit of sign X to the 5th race to the 6th divided by X the 4th as X approaches infinity. So let's go ahead and try to find that. Now, the limit, as X approaches infinity. OK, the limit as X approaches infinity. Of 0 is going to be equal to 0. And the limit as x approaches infinity of 1 divided by X to the 4th is also going to be equal to 0 because like we did for 35 divided by X as X gets arbitrarily large, our term tends to 0, OK? Since both of them are equal to 0, OK, that must mean the limit of the term in between. The sign of x to the 5th to the 6th divided by 4 as x approaches infinity will also be equal to 0. Therefore, if we go back to our problem, if we finish it, that means this limit that we just found is also going to be equal to 0 and 12 + 0 + 0 equals 12. Thus, the limit as X approaches infinity of our function is 12. C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Determine the following limits.
lim x→−∞ (5 + 100/x + sin⁴ x³ / x²)

Key Concepts

Limits at Infinity

Dominant Terms

Trigonometric Limits

Watch next