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) = (3x⁴ + 3x³ − 36x²) / (x⁴ − 25x² + 144)

Accepted Answer

Welcome back, everyone. Evaluate the limit as x approaches positive and negative infinity and identify any horizontal asymptotes for the function F of X is equal to 2x the power of 4th plus 6 X cubed minus 20 x 2 divided by x to the power of 4 minus 29 x2 + 100. So let's evaluate the two limits. The first one is going to be limit as X approaches positive infinity. We're going to rewrite our function. So we are including the numerator and now the denominator. What we're going to do is attempt the right substitution. Let's assume that x approaches infinity, which is given in this case. In the numerator we're going to get infinity minus infinity. In the denominator we're going to get infinity minus infinity, and this is an indeterminate form. But since it's a rational function and X approaches infinity, what we have to do is basically divide both sides of the given fraction by X with the highest exponent in the denominator, and that's X to the power of 4th. So if we divide each term by x to the power of 4th, we're going to get 2 in the numerator, plus 6 x cubed divided by X to the power of 4th, which gives us 6 divided by X minus 20 x 2 divided by X4, which gives us 2. T divided by x 2 and in the denominator we're going to get 1 minus 29 divided by x2 + 100 divided by X to the power of 4th. Now x approaches infinity, so every fractional term approaches 0 because we have a number divided by X. To the power of 1st, 2nd, and 4th. So basically whenever this is the case and x approaches infinity, the ratio approaches 0, and we simply get 2 divided by 1, which is 2. Now when we consider the limit as x approaches negative infinity. We can use the. Warm that we had previously right. Which was our rearranged form, and now we have to understand that nothing really changes. We have two whole numbers and then we simply have those zeros from the left or from the right. It doesn't really matter. The limit is still 0, right? So it doesn't affect the ratio of 2 to 1, and we still have the same limit of 2. So we understand that there is only one horizontal asymptote in the form of Y equals 2. Right? Because the two limits give us the horizontal asymptotes. So the first limit is equal to 2, the second one is equal to 2 as well, and the horizontal asymptote has a form of Y equals 2. Thank you for watching.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x) = (3x⁴ + 3x³ − 36x²) / (x⁴ − 25x² + 144)

Key Concepts

Limits at Infinity

Horizontal Asymptotes

Polynomial Functions

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