1. V ERTICAL AND H ORIZONTAL A SYMPTOTES

2. V ERTICAL A SYMPTOTES To find Vertical Asymptotes: Step1. Factor the top and bottom of the function completely. Step2. See if anything “mentally” cancels top to bottom. Step3. The vertical asymptotes are the values of x that make the bottom zero after you’ve “cancelled”.

3. Therefore, the vertical asymptotes for a function are the values of x that make the bottom zero, BUT not the top. In a graph, the function will NEVER cross a vertical asymptote. It approaches the vertical asymptote on the left and will either go to + ∞ (up) or - ∞ (down). The same things happens as the function approaches the vertical asymptote from the right side. Facts to Remember

4. Example: This function cannot be factored. The value of x that makes the bottom zero is x = -1. (Notice the x = -1 makes the bottom zero, but not the top). So, x = -1 is the vertical asymptote. Look at the picture below and you will see how the graph behaves on each side of x = -1. The vertical asymptote is an imaginary vertical line that the graph cannot cross.

5. Example: Notice the function has been factored. Mentally cancel out the x - 4 and that leaves x + 3 As the factor that can be zero in the bottom. Therefore, x = -3 is the only vertical asymptote. ***NOTE: When finding the domain, both 4 and – 3 would have to be left out. You would NOT mentally cancel out the x – 4 to find domains. Also, 4 and – 3 are both discontinuities as well. But x = - 3 is the only vertical asymptote. We would say that x = 4 is a removable discontinuity, not a vertical asymptote.

6. H ORIZONTAL ASYMPTOTES To find Horizontal Asymptotes: Step1. Do not factor!!! Compare the highest power of the variable top to bottom. Step2. One of three things will be observed. 1. If the highest power of x in the top is bigger than the highest power in the bottom, there is no horizontal asymptote. 2. If the highest power of x in the top is smaller than the highest power in the bottom, the horizontal asymptote is y = 0. 3. If the highest power of x on top and bottom is equal then horizontal asymptote is y = the fraction formed by the coefficients of the highest power of the variable in top and bottom.

7. F ACT TO R EMEMBER : A horizontal asymptote is a dotted horizontal line in the graph of the function that “draws” the function at the “ends” of the graph. In other words, these horizontal lines act like magnets for the function as x goes toward - ∞ or + ∞. Horizontal asymptotes can be crossed, but seldom are.

8. Case 1: has no horizontal asymptotes at all. Why? The highest power of x in the top (which is 3) is bigger than the highest power or x in the bottom (which is 2). So, as x gets bigger and bigger (closer and closer to ∞) the values in the top of the fraction are growing much faster than the values of x in the bottom of the fraction. Therefore, the fraction itself grows bigger and bigger and does NOT approach any particular number value (y – value). The same could be said as x gets smaller and smaller (closer and closer to - ∞). So there is no y – value that the function approaches.

9. Case 2: has a horizontal asymptote of y = 0. Why? This function is an example where the highest power of x in the top (which is zero) is smaller than the highest power of x in the bottom (which is 1). Therefore as x gets bigger and bigger (closer and closer to ∞) the bottom of the fraction gets bigger and bigger. However, that means that the fraction gets tiny (very, very close to zero). The same could be said as x gets more and more negative (closer and closer to -∞). Therefore, case 2 says the horizontal asymptote would be: y = 0. In its graph, the function is drawn to the imaginary horizontal line y = 0 at the far left and at the far right of the picture.

10. Case 3: has a horizontal asymptote of. Why? Notice that the highest power of x in the top and bottom of the fraction is 2 (the same). So, as x gets bigger (closer and closer to ∞) the values in the top and bottom both grow bigger. However, together they look more and more like. For instance, when x = 1000, the fraction is which is very close to. Similarly, when x = -1000, the fraction is also very close to as well.