How to find Horizontal Asymptotes?

A horizontal asymptote for a function is a horizontal line that the graph of the function approaches as x approaches ∞ () or -∞ (minus infinity). In other words, if y = k is a horizontal asymptote for the function y = f(x), then the values (y-coordinates) of f(x) get closer and closer to k as you trace the curve to the right (x→ ∞) or to the left (x → -∞). Because asymptotes are defined in this way, it should come as no surprise that limits make an appearance. The precise definition of a horizontal asymptote goes as follows: We say that y = k is a horizontal asymptote for the function y = f(x) if either of the two limit statements are true: Limit definition of asymptotes.

A function can have two, one, or no asymptotes. For example, the graph shown below has two horizontal asymptotes, y = 2 (as x→ -∞), and y = -3 (as x→ ∞).

If a graph is given, then simply look at the left side and the right side. If it appears that the curve levels off, then just locate the y-coordinate to which the curve seems to be approaching. It helps to sketch a horizontal line at the height where you think the asymptote should be. Let’s see how this works in the next example. Keep in mind, you will typically not be shown the dashed line — that would make the problem way too easy!

The graph on the left shows a typical function. If you follow the left part of the curve as far to the left as you can, where do you end up? In other words, what is the y-coordinate of the leftmost point shown in the graph? A good estimate might be somewhere between 1 and 2, perhaps a little closer to 1.

Well imagine what would happen if you continued drawing the graph to the left of what is shown. It seems reasonable that the curve levels off and approaches a value of 1, gently touching down on the horizontal line y = 1 just like an airplane landing.

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