Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any ...

Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=|1−x^2| / x(x+1)

Accepted Answer

Welcome back everyone. Determine the horizontal asymptotes of the function J X equals the absolute value of 4 minus X2 divided by X multiplied by 3 + 2 X. And were given for answer choices A says Y equals -12 and 1/2 B says Y equals -12 C Y equals 12, and D Y equals 2. So if we want to identify the horizontal asymptotes of a function, we want to evaluate two limits. The first one is going to be limit when X approaches negative infinity of the function, and the second one is going to be the limit when X approaches positive infinity. So let's substitute for J of X, and that's the absolute value of 4 minus X2 divided by. X multiplied by 3 + 2 X. Now since we're given the absolute value, we have to understand that it is a piecewise function. When we square x, well, negative infinity is going to become positive infinity, and if we subtract 4 minus an infinitely large number, we're going to get a negative value inside of the absolute value. And let's remember that whenever we take the absolute value of X. It is going to be equal to negative X if X is less than 0, or X, if X is greater than or equal to 0. So for this case, since we're going to get a negative value within the absolute value. Well, we can essentially. At a negative sign in front of the numerator and drop the absolute value sign. So we get limit as x approaches negative infinity. Of negative. Or minus X2. And in the denominator, we can distribute the terms, so we get 2 X2. Plus 3 X. Now from here, if we apply the direct substitution, well, we're going to get infinity divided by infinity, that's an indeterminate form. So To simplify it, we're going to divide both sides of the given rational function by X2 because that's the variable with the highest exponent. First of all, let's factor out the negative sign, and then we get negative limit as X approaches negative infinity. And now let's divide all of our terms by X2. So we get 4 divided by X2 minus. X2 divided by X2. In the denominator, we get 2 X 2 divided by X2 + 3 X divided by X2, and then when we simplify, we get negative limit. S X approaches negative infinity. Of Or divided by X2 minus 1. Divided by 2 + 3 divided by X. Now it doesn't really matter if X approaches negative or positive infinity. 4 divided by X squared and 3 divided by X, those will approach 0, right? Because we have a number divided by infinity. And we can basically say that the two limits are going to be equal to each other, so we can essentially finish this problem by evaluating just this one limit, because it will give us the same answers we get positive or negative infinity for each limit. And let's add that positive sign for X approaches negative infinity. And now let's evaluate the limit. We get negative. In the numerator we get -1. In the denominator we have 2. So negative of -12 gives us positive 12, and that's the expression of the horizontal asymptote Y equals 12. So looking at the answer choices, we can conclude that the correct answer is option C. Thank you for watching.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=|1−x^2| / x(x+1)

Key Concepts

Limits at Infinity

Horizontal Asymptotes

Absolute Value Functions

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