Horizontal asymptotes Determine limx →∞ f(x) and limx →-∞ f(x) for the following functions. Then

Horizontal asymptotes Determine limx →∞ f(x) and limx →-∞...

Horizontal asymptotes Determine $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for the following functions. Then give the horizontal asymptotes of $f(\text {if any})$.
$$f(x)=\frac{3 x^{3}-7}{x^{4}+5 x^{2}}$$

Key Concepts

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches infinity or negative infinity. It reflects the limit of the function at the extreme ends of the x-axis, indicating the behavior of the function far from the origin.

Degree Comparison Test

The Degree Comparison Test is a technique used to determine the end behavior of a rational function. By comparing the highest degrees of the numerator and denominator, one can ascertain whether the limit as x tends to infinity or negative infinity will be zero, a finite constant, or if the function will diverge.

Limits at Infinity

This concept involves analyzing the behavior of a function as the input variable increases or decreases without bound. By evaluating limits as x approaches infinity or negative infinity, we determine the end behavior of functions, which is crucial in understanding asymptotic tendencies.

Rational Functions

Rational functions are expressions defined by the ratio of two polynomials. Their behavior at large values of x often depends on the degrees of these polynomials, and studying them helps identify key characteristics such as asymptotes and intercepts.

Transcript

00:03 We have a rational function here, and we want to determine the end behavior.

00:08 In other words, what happens to this function as x becomes very large in the negative direction and as x becomes very large in the positive direction? to determine that, re -expressing this function will be handy.

00:28 So let's factor out x to the fourth from the numerator and the denominator.

00:32 Or 3x cubed divided by x to the fourth gives us 3 over x.

00:38 7 divided by x to the fourth is 7 over x to the fourth.

00:43 And in our denominator, we'll do the same thing.

00:46 We'll factor out x to the fourth.

00:48 So we end up with x to the fourth over x to the fourth is one.

00:52 And 5x squared over x to the fourth gives us 5 over x squared.

00:59 Now, these x to the fourths can cancel.

01:04 And what we're left with is fractions that have the x's in the denominator, we're also left with this constant here.

01:12 So no matter what x is, one is one.

01:15 But as x changes, these other terms in our function change.

01:19 So we're going to make ourselves a little table and determine what happens to those things.