To begin this topic, we first define all types of asymptotes.
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.
A function f is said to have a horizontal asymptote at y = b if
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or if
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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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Try it nowHere'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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It is clear from these last two observations that x = 0 is a vertical asymptote. The plot of this function is shown in Figure 2.
Figure 2:
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Now here's our third example: Find all asymptotes for the function:
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Again, we have to sketch using a graphing calculator.
And here's our solution:
Let's begin by noticing that x = 2 is a candidate for a vertical asymptote. Evaluating the one-sided limits there, we see that:
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and
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Now note that we can rewrite the function as (completion of square):
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This implies upon substitution:
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Therefore y = x + 2 is a linear asymptote, and we would have the same result if we let x tend to negative infinity. (Same argument.)
A sketch of the graph is shown in Figure 3. Included on the graph is the plot of y = x + 2. The function's graph becomes closer to the line y = x + 2 as x tends to positive and negative infinity.
Figure 3:
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Let's take a couple of moments to recap the essential information about asymptotes and how to find them that we learned in this lesson. An asymptote for a function indicates that the function is close to linear as x approaches a finite number or positive or negative infinity. Asymptotes can be horizontal, vertical, or a line pointed in any direction.
If the denominator for a function is zero at a point, then it's possible that the function has a vertical asymptote at that point. Finding a horizontal asymptote amounts to evaluating the limit of the function as x approaches positive or negative infinity.
Finding a linear asymptote may require a bit more algebra, since it's along the line y = ax + b, although the first clue isn't nearly as complicated. For instance in Example 3, the numerator is a quadratic function, and the denominator is a linear function, so the difference in degrees is one, suggesting that for large x, we are basically looking at a polynomial of degree one (a linear function). We could have had a fourth degree polynomial divided by a third degree polynomial, and then we'd be seeking the same type of asymptote. Only the algebraic work would differ.
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