**Limits at Infinity**

This lesson will serve as a preparation for the graphing in the next lesson. Here, we will work on ways to identify asymptotes from the formula of a rational function. *Rational functions* are quotients, with a clear numerator and denominator.

*Vertical asymptotes* are easy to recognize, because they occur where the denominator is undefined. For example, has vertical asymptotes at *x* = – 3 and *x* = 4.

*Horizontal asymptotes* take a bit more work to identify. The graph will flatten out like a horizontal line if large values of *x* all have essentially the same *y*-value.

In this graph of *y* = *f*(*x*), for example, if *x* is bigger than 5, then *y* will be very close to *y* = 1 (see Figure 13.1). Thus, *y* = 1 is a horizontal asymptote. Similarly, if *x* is a large negative number, the corresponding *y*-value will be close to zero. Thus, *y* = 0 is another horizontal asymptote. Horizontal asymptotes are related to the limits as *x* gets really big. For *f*(*x*) given in the graph:

**Asymptote Hint**

**Asymptote Hint**

*Notice that the graph of y = f(x) crosses both asymptotes. Vertical asymptotes cannot be crossed because they are, by definition, not in the domain. Horizontal asymptotes can be crossed, as illustrated in this example. Think of "asymptote" as meaning "flattens out like a straight line" and not "a line not to be crossed."*

These *limits at infinity* (and negative infinity) identify what the ends of the graph do. For example, if , then the graph of *y* = *g*(*x*) will look something like that in Figure 13.2. If , then the graph of *y* = *h*(*x*) will look like that in Figure 13.3.

Notice that the infinite limits say only what happens way off to the left and to the right. Other calculations must be done to know what happens in the middle of the graph.

The general trick to evaluating an infinite limit is to focus on the most powerful part of the function. Take , for example.

There are a lot of negative elements to this function. However, the most powerful part is the positive 2*x*^{3}.When *x* gets big enough, like when *x* = 1,000,000, then 2*x*^{3} – 100*x*^{2} – 10*x* – 5,000 = 2,000,000,000,000,000,000 – 100,000,000,000,000 – 10,000,000 – 5,000 = 1,999,899,999,989,995,000

This clearly rounds to 2,000,000,000, 000,000,000, which is the 2*x*^{3}. It is in this sense that 2*x*^{3} is called the most powerful part of the function. As *x* gets big, 2*x*^{3} is the only part that counts.

**Rules for Infinite Limits**

**Rules for Infinite Limits**

*The rules for Infinite Limits of Rational Functions are as follows:*

*If the numerator is more powerful, the limit goes to ∞ or –∞.**If the denominator is more powerful, the limit goes to 0.**If the numerator and denominator are evenly matched, the limit is formed by the coefficients of the most powerful parts.*

As *x* gets huge, *x*^{3} is clearly even larger, and 2*x*^{3} is twice that. Thus, as *x* goes to infinity, so does 2*x*^{3}. Basically, the higher the exponent of *x*, the more powerful it is. With that in mind, the rules for infinite limits of rational functions are fairly simple:

- If the numerator is more powerful, the limit goes to ∞ or –∞.
- If the denominator is more powerful, the limit goes to 0.
- If the numerator and denominator are evenly matched, the limit is formed by the coefficients of the most powerful parts.

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