Horizontal and Vertical Asymptotes

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Question

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all horizontal asymptotes.

_____

√x² + 4

c. g(x) = -----------

x

Accepted Answer

Welcome back, everyone. Identify the horizontal asymptote for the function F of X equals square root of 4 X2 + 1 divided by 2 x using limits. We're given 4 answer choices A says Y equals 0 and Y equals -1. B Y equals -1 and 1. C Y equals -1, and D Y equals 0 and 1. So if we want to identify the horizontal asymptotes of a function, we have to identify two limits. Let's begin with the limit as X approach is positive infinity, and then we're going to consider the same limit as X approaches negative infinity. We're going to begin with positive infinity. We're going to substitute our function, which is square root of 4 X2 + 1 divided by 2 X. If we simply at time direct substitution, this is going to give us infinity divided by infinity, and if we have a rational function, the best idea is to divide both sides of our fraction by X with the highest exponent, and this is what we're going to do. But since we have a radical term, we're also going to factor out X2. So let's begin with the numerator and let's factor out X2d. Which gives us Or plus 1 divided by X where the parent is our factor and then in the denominator we still have 2 X. Now let's remember that we can take square root of X2, which is the absolute value of X, so we get limit as X approaches infinity of the absolute value of X multiplied by square root of 4 + 1 divided by X2, and then in the denominator we still have 2 X. So now if X approaches positive infinity, we can simply drop the absolute value, right, because X is positive. And as we can see, we can basically cancel out the axis. And now 1 divided by X2d approach is 0 because we have a number divided by infinity, so we can say that. This term approaches 0, and this allows us to evaluate the result. In the numerator we have square root of 4 + 0, which is square root of 4. In the denominator we have 2. So square of 4 divided by 2 is simply 1, meaning we have identified our first horizontal asymptote, which is Y equals 1. And now we're going to perform the same analysis. When X. Approaches negative infinity. And we're going to take our previously defined function, so it's the absolute value of X multiplied by square root of 4 + 1 divided by X2, all of that divided by 2 X. And now because X approaches negative infinity when we drop the absolute value since the term inside is negative, well, we need to add a negative sign in front of X. This is really important according to the definition of the absolute value function, which is a piecewise function, but now we can see again that we can cancel out those axes. The only difference is that we get a negative sign. So for our answer, we're going to get -1 in this case due to the presence of the negative sign, which means that the second horizontal asymptoto Y equals -1. Therefore, we have two horizontal asymptotes Y equals 1 and Y equals -1, which corresponds to the answer choice b. Thank you for watching.

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all horizontal asymptotes.
_____
√x² + 4
c. g(x) = -----------
x

Key Concepts

Limits

Horizontal Asymptotes

Rational Functions

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