Determine the following limits at infinity.

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Question

Determine the following limits at infinity.

lim x→∞ (5 + 1/x +10/x^2)

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the limit of the given function at infinity. The limit as X approaches infinity of 113 + 5 divided by X plus 101 divided by X to the power of 2. Awesome. So it appears for this particular prom we're trying to figure out what the limit is of this given function at infinity. So now that we know that we're trying to figure out this limit of the given function at infinity, let's read off our multiple choice sensors to see what our final answer might be. A is -113. B is negative infinity. C is 113, and D is 226. OK. So first off, in order to find the limit of the function, which we'll call our function F of X, and we'll say that F of X is equal to parenthesis 113, and then we need to take 113, and we need to add 5 divided by X, and then we need to add another fraction, you need to add 100. In 1 divided by X to the power of 2. Fantastic. And once again, this expression is given to us in the problem itself as limit as X approaches infinity. OK, so, as we should recall a note, we're trying to find the limit of this function as X approaches infinity. And in order to do that, we can follow 4 simple steps to solve this problem. So in 4 easy steps we'll be able to solve this problem quick and easy. So our first step is we need to recall and note that the function has 3 terms. It has a constant term which we'll call CT, and our constant term, which I'll highlight in blue here is our 113, so our constant term is 113. And we also have two fractions. We have our fraction, where I'll say F is 5 divided by X, and then we also have another fraction, which is 101 divided by X squared. So let's make sure we put an F there. So we have an F for fractions. So we have two fractions and a constant term for this particular problem. So, As we should recall now, our second step, we need to note that this constant term of 113 will remain unchanged regardless of the value of X. So, so 113 will be unchanged, which we'll call UC unchanged, regardless of the value of X. And we also need to recall a note that the term 5 divided by X approaches 0, so let's make a note of this. So we need to note that as 5 divided by X approaches 0. As X approaches infinity, and it's important to note that we're saying positive infinity. And this is the case because the numerator is a constant term, while the denominator grows without bound. And similarly, we need to note that For other fraction, we could say that 101. Divided by X to the power of 2, this one will also approach 0 as X approaches infinity. And this is for the same reason. So both of our fractions, so both of our fractions will be approaching zero, because the numerator will is a constant term while the like by the denominator, which is the value below the division bar, so the bottom number, that's our denominator. So, It's so both of these fractions approach 0 because we have a constant term on the top part of our division bar, and then we have our X to the power of 2 term, which will be growing without bound, and this is the same case for both of our fractions. Awesome. So now we can move on to our third step. So our third step we need to recall and note that since the only term that does not approach zero is the constant term 113, the limit of the entire function as X approaches infinity will be the same will be simply the value of this constant. So that means that as X approaches infinity, the limit. So the limit will equal 113. Or rather, the find the limit of the given function at infinity. So that means when we go to our 4th and final step, we can safely recall and write that the limit as X approaches infinity. Of 113 + 5 divided by X plus 101 divided by X squared. will be all equal to 113, and this is our final answer. Hooray, we did it. And this corresponds to multiple choice answer letter C, which once again is 113. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Determine the following limits at infinity.

lim x→∞ (5 + 1/x +10/x^2)

Key Concepts

Limits at Infinity

Dominant Terms

Continuous Functions

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