Finding Asymptotes Using Limits

Lesson Transcript
Instructor: Christopher Haines
A function can have a vertical asymptote, a horizontal asymptote and more generally, an asymptote along any given line (e.g., y = x). In this lesson, we learn how to find all asymptotes by evaluating the limits of a function.

What are Asymptotes?

To begin this topic, we first define all types of asymptotes.

Definition 1: Vertical Asymptote

A function f is said to have a vertical asymptote at x = a if


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or if


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These limits can be one-sided or two-sided.

Definition 2: Horizontal Asymptote

A function f is said to have a horizontal asymptote at y = b if


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or if


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Definition 3: Linear Asymptote

A function f is said to have a linear asymptote along the line y = ax + b if


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or if


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A horizontal asymptote is a special case of a linear asymptote. This is the case of a = 0.

So, as we've seen with these three types of asymptotes, the concept of an asymptote (whether vertical, horizontal, or linear) is a function moving along a line in the limit as x approaches a finite number or positive or negative infinity. In the examples that follow, we're going to use the method of limits to locate asymptotes of various functions.

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Finding a Function's Asymptotes

Here's our first example: Find all asymptotes for the function:


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You can sketch the function by using a graphing calculator.

Now here's our solution:

By dividing the numerator and denominator by x, we can evaluate the limit as x approaches infinity:


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Similarly,


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In accordance with Definition 3, f has a horizontal asymptote at y = 0, and it has it in both directions towards positive and negative infinity.

Now note that by factoring the denominator, we can rewrite the function as:


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With the function expressed this way, it becomes apparent that there are two suspects for vertical asymptotes. This is because x = -1 and x = 1 give a zero in the denominator. So we investigate these two points further and also be mindful of which side we are approaching each of the two points.

The one-sided limits are as follows. When x < -1, all three of the factors (numerator and denominator combined) are negative. So therefore:


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When -1 < x < 0, two of the factors are negative, and one of the factors is positive, so the function is positive on this interval, and the right-sided limit at x = -1 is


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Similarly for the other candidate,


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and


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The graph of the function is shown in Figure 1. The asymptotes in the plot are consistent with our calculations. However, sketching the plot in its entirety (without a graphing utility) is outside the scope of this lesson.

Figure 1:


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Here is our second example: Find all asymptotes for the function:


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Again, let's sketch the graph using a graphing calculator.

Now here's our solution:

Note that this function is really the same as


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We can obtain the latter expression upon multiplication on the numerator and denominator by exp(x). Clearly,


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and


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So the function has two horizontal asymptotes: one for each direction of positive and negative infinity. They are y = 0 and y = -1.

Since the denominator is zero when x = 0, the only candidate for a vertical asymptote is x = 0. We will need to consider both one-sided limits as x approaches zero.

When x < 0, exp(x) < 1, so the left sided limit is then:


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And, similarly, when x > 0, exp(x) > 1. So the right sided limit is:


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