Essential Knowledge 1.1D1

Essential Knowledge 1.1D1 Students will know that asymptotic and unbounded behavior of functions can be explained and described using limits.

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Asymptotes

Asymptotes are lines that show where a curve will approach, but never quite intersect (except theoretically at infinity). A picture will show this idea more clearly.

Let's look at a graph of

This curve seems to have some boundaries on it. Parts of the curve are flattening out into straight lines. If the curve extended infinitely far, it would become a straight line. We can draw asymptotes to show where these straight lines would be.

As x → ∞, y→1 and the curve approaches the line y = 1, but never quite touches it (except theoretically at x = ∞). The blue line, y = 1, shows this, and we call this line a horizontal asymptote. The table below also shows this happening.

As x → 10, y→ -∞ and the curve approaches the line x = 10, but never quite touches it (except theoretically at x = 10). The green line, x = 10, shows this, and we call this line a vertical asymptote. The table below also shows this happening.

Thus far we've made estimates visually and using tables. Of course, these guesses seem really good and so they probably are correct. However, to make our answers official we must solve using limit theorems and algebraic rules. So lets think about how we could find these asymptotes without estimating.

Horizontal Asymptotes

I noticed that x→∞ in order to find my horizontal asymptote. This is key. In fact, this is exactly how you find horizontal asymptotes. Since the curve would need to extend infinitely far to become a flat horizontal line, see what happens when x → ∞ or -∞. If you get a real number, for example 5, then y = 5 will be your horizontal asymptote.

Vertical Asymptotes

I noticed that y→ -∞ in order to find my vertical asymptote. This is key. In fact, this is exactly how you find vertical asymptotes. Since the curve would need to extend infinitely far to become a straight vertical line, see what x values make y → ∞ or -∞. If you can find a real number, for example -5, then x = -5 will be your vertical asymptote.

Example 1

Looks like our only horizontal asymptote will be y = 2

Now let's find the vertical asymptotes. We need to find values that make y → ∞ or -∞. Well, we let's think back to what we learned in Essential Knowledge 1.1CP for making answers infinite.

• We could make a fraction infinite if we made the numerator infinite. Well, this won't work because only the values x→∞ and x→∞ would make 2x² infinite.

• We could make an indeterminate fraction of 0/0, which might infinite once we make it determinate. Well, this won't work because only x = 0 makes the numerator 0, and the denominator would be -3

• Our last hope is to make the denominator 0, let's try it.

So, what values will make x² - 2x - 3 equal to zero? Hopefully you remember your factoring!

x² - 2x - 3 = 0

(x - 3)(x + 1) = 0

x = 3, x = -1

We don't need to, but let's check our work.

(I'm taking a bit of a shortcut here, if you want to know why I'll explain at the end.)

Indeed these x values do cause y to approach infinity, so we can say our curve has vertical asymptotes at x = 3 and x = -1.

Let's also check our work by graphing. As you'll see, our answers work perfectly.

Note about vertical asymptotes

Hopefully you noticed that my limits were incorrect for the vertical asymptotes. For example x→-1. Looking at the graph, clearly the limit isn't really ∞. From the left the limit approaches ∞, but from the right the limit approaches -∞. So if I wanted to be technically correct, I would have had to do one-sided limits.

So why didn't I do this? Because it really doesn't matter. If you are dividing a real non-zero number by 0, your limit will either be

• ∞

• -∞

• ∞ from one side and -∞ from the other

And the bottom line is, all three situations still have a horizontal asymptote at that value of x. So don't worry. You just make that denominator zero and you'll be fine.