Determine the following limits at infinity.

l...

Question

Determine the following limits at infinity.

lim x→∞ (3+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 5 + 101 divided by X to the power of 4. Awesome. So it appears for this particular problem we're ultimately trying to figure out the limit of the given function at infinity. Awesome. So with that said, let's read off our multiple choice answers to see what our final answer might be. A is negative infinity, B is -5. C is 5, and D is positive infinity. Awesome. So, first off, we need to find the limit of the function. So let us write that F of X, so our function F of X. It's going to be equal to parenthesis 5 + 101, and we need to take 101, and then we divide it by X to the power of 4, and then don't forget your second parenthesis. And we're trying to find the limit of this function as X approaches infinity. Awesome. So in order to do that, we can solve for this problem in 4 simple steps. So let us solve for this problem in 4 simple steps. Awesome. So our first step is we need to recall and note that this function is composed of a constant term which we'll call CT, and our constant term is 5, as we could see in our problem itself. And then we also have a fraction which we'll just call F, and the fraction is equal to 101 divided by X to the power of 4. For our second step, we need to recall and note that this constant number 5 or this constant variable 5 will remain unchanged regardless of the value of X. We also need to note that for 101 divided by X to the power of 4, as this approaches 0. We need to note that So we need to note that the term 101 divided by X the power of 4 as this approach is 0, this is happening as X approaches infinity because the denominator X the power of 4 will be growing exponentially, making the function's value decrease towards 0. So once again, as 101 divided by X to the power of 4, as our fraction approaches 0. This is happening as X as X is approaching infinity. And this is because it's due to the fact that the this denominator, which is X to the power of 4, is growing, which makes our fractions value decrease towards 0. So I'll do a capital D with the downwards arrow, so it's decreasing towards 0. So decreasing down towards 0. Awesome. So our third step is we need to note that once again, as we should recall a note that since this constant variable 5 remains 5 and our fraction 101 divided by X to the power of 4 approaches 0, the sum of these two as X approaches infinity is 5 + 0, which will be equal to 5. So now we can officially write our limit as our 4th and last step. So therefore the limit. As X approaches infinity, so we could say that the limit as X approaches infinity of 5 + 101 divided by X to the power of 4 will be equal to 0. So, the limit of the function as X approaches infinity, so this is as X approaches infinity, that will mean that the limit is 5 and 5 is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be multiple choice answer letter C, which once again is 5. 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→∞ (3+10/x^2)

Key Concepts

Limits at Infinity

Rational Functions

Dominant Terms

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