Horizontal and Vertical Asymptotes

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Question

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all horizontal asymptotes.

_________

/ x² + 9

d. y = / -------------

√ 9x² + 1

Accepted Answer

Welcome back, everyone. Identify the horizontal asymptotes for the function F of X using limits. Were given 4 answer choices A says Y equals 25 and square root of 5 B Y equals 25. C Y equals square root of 5, and D Y equals square root of 5 divided by 5. So we're given our F of X and to identify the horizontal asymptotes, we want to evaluate two limits. Limit as X approaches positive or negative infinity of F of X, and our F of X is square root of 5 X2 + 16. Divided by x2 + 1. Now what we notice is that if we substitute x approaches infinity for every X, we're going to get square root of infinity divided by infinity, which is an indeterminate form. Well, we can basically rewrite it as infinity divided by infinity. And since this is an indeterminate form, we want to Essentially perform modifications for our function to get a different expression and see if it solves the problem and we end up with a finite value, right? So what we're going to do in this case, since we have infinity divided by infinity and X approaches infinity, we have to notice that we have a rational function under the radical. So the rule says that what we want to do is simply divide the numerator and the denominator by the X with the highest exponent. And that's x 2. So we're going to divide all of our terms by x2, which gives us 5 + 16 divided by X2 in the numerator, and for the denominator we get 1 + 1 divided by X 2 in the denominator. And now what we notice is that as x approaches infinity, 16 divided by X2d approach is 0 and 1 divided by X2d approaches 0 as well because we have a number divided by an infinitely large number. So we're only left with square root of. 5 divided by 1, which is square root of 5, and this is how we get our first horizontal asymptote in the form of Y equals square root of 5. Now for the second horizontal asymptote we're going to evaluate the limit as X approaches negative infinity, right, so we have to check the positive infinity and the negative infinity. In here we can use our previously obtained function, which is square root of 5 + 16 divided by X2, all of that divided by 1 + 1 divided by X2, right? Now, if we have a number divided by infinity, well, in any case, we get 0, regardless of whether it's negative or positive infinity. So 16 divided by X squad. Would still be 16 divided by negative infinity squared which is positive infinity, right, and 1 divided by negative infinity squared also gives us 1 divided by positive infinity because when we square a negative number, it becomes positive. However, if we simply have a number divided by positive or negative infinity, the limit is 0. So in this case we also get 0 for each fractional term. And essentially we get the same value for the limit square root of 5. Divided by one. And this is again square root of 5 equals Y. So since those two results are the same, it just means that we have one possible horizontal asymptote in the form of Y equals square root of 5, and our final answer corresponds to the answer choice C. Thank you for watching.

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all horizontal asymptotes.
_________
/ x² + 9
d. y = / -------------
√ 9x² + 1

Key Concepts

Limits

Horizontal Asymptotes

Rational Functions

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