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## Algebra (all content)

### Unit 13: Lesson 8

End behavior of rational functions

# End behavior of rational functions

Sal analyzes the end behavior of several rational functions, that together cover all cases types of end behavior.

## Want to join the conversation?

• Can anyone explain to me the logic behind "approaching from below" and "approaching from above". I've been doing the exercises and I don't understand the explanation given there. How do I know from where the graph will approach 0? •  100 -> 10 -> 1 -> .1 -> .01 is approaching 0 from above, or from the positive (positive numbers are 'above' 0)
-100 -> -10 -> -1 -> -.1 -> -.01 is approaching 0 from below, or from the negative (negative numbers are 'below' 0)

As x approaches infinity (as x gets bigger):
1/x approaches 0 from above (smaller and smaller positive values)
-1/x approaches 0 from below (smaller and smaller negative values)
• In the first equation, Sal simplifies the equation to: (7x - 2) / (15 - 5/x). He states at that 7x will approach negative infinity, however he simplified this value from 7x^2 which by definition will always turn negative numbers positive. Shouldn't he have simplified the equation to state: 7|x|? •  Sal simplified the expression down to it's dominate terms: 7x/15.
If you are going to use the 7x^2, then you also need to use the 15x in the denominator. Yes, 7x^2 would be positive, but 15x would be negative. A positive / a negative = a negative.
Hope this helps.
• Can someone please help me because I am still very confused. How does Sal know that as the numbers approach - infinity they will get to 0 and why is this? What happens as it approaches infinity and why? • Something approach positive infinity means that it become larger and larger, extremely large. And if something approach negative infinity means it becomes smaller and smaller, as small as you can imagine, and even smaller than that. And if, for example, some number infinity approach 3, means it is so close to 3, extremely close but not yet 3. So, of course, it's not a real number, the symbol of infinity is 8 turn 90 degrees.

And if you want to know which type of approach the question is, you could plug some value in x to figure out what does it approach.

Example 1: What does 7x-2 approach if x approach negative infinity?

What this question means is what number is 7x-2 approach if x become extremely small.
1. If x is -1, 7x-2 is -9
2. If x is -10, 7x-2 is -72
3. If x is -100, 7x-2 = -702
Here's a pattern, as x become smaller and smaller, 7x-2 become smaller and smaller as well. That means when x approach negative infinity, 7x-2 approach negative infinity as well.

Example 2: What does 6x^5/x^9 approach if x approach infinity?

So, this question is similar to the first one, but this question doesn't tell us x is approach to positive infinity or negative infinity. But lucky for us, we don't need to know.
1. If x is 100, 6x^5 is 7.776×10^13, x^9 is 1×10^18, answer is 7.776×10^-5 (it's a very small positive number, but not yet zero)
2. If x is 10, 6x^5 is 777600000, x^9 is 1000000000, answer is 0.7776
3. If x is -10, 6x^5 is -1.29×10^-9, x^9 is -1000000000, answer is 1.29×10^−18
4. If x is -100, 6x^5 is -1.29×10^-14, x^9 is -1×10^18, answer is 1.29×10^-32 (it's a very small positive number, but not yet zero)
So as you can see again, when x become extremely larger or extremely smaller, 6x^5/x^9 extremely approach to 0, but not yet 0.

Okay, hope that helps! :)
• Couldn't you plug in +infinity and -infinity and decide? • How did Sal know that dividing the numerator and denominator of the function by the highest degree of "x" found in the denominator would make the entire function easier to analyze? In other words, how does he know that the method used in the video will make the function easier to analyze? • I'm still trying to figure this video out. From reading other comments it seems that dividing by the highest degree in the denominator you make the denominator like a constant as the variable goes to positive or negative infinity. That means that you can concentrate on what is happening in the numerator. I guess whoever came up with it was just trying "algebraic changes which do not change the result" and which could make analysis easier (like using -1.-1 or multiplying by a crazy denominator over that same crazy numerator - those examples both multiply by one but can change the algebra) .
• • So all vertical asymptotes are discontinuities, but not all discontinuities are vertical asymptotes, right? • • there is question in the practice 6x^4-7x/3x^7+18x when x approach postive infinity
what the function approach,i used sal's method, using the highest degree to figure it
out, but the question asked is it approaching 0 from above or below,how would i know is it approaching from above or below? • If f(x) is the function, then as x approaches infinity f(x) approaches 0 from above.
You have established that there is a horizontal asymptote at y=0 [6x^4 / 3x^7 approaches 0 as x is large].
To determine whether f(x) approaches the asymptote from above or below consider the sign of f(x) as x is large.
f(x) = (6x^4 -7x) / (3x^7 +18x)
When x is large both the numerator and denomenator will have +ve signs giving f(x) an overall +ve sign, so f(x) must approach from above.
If the question had asked whether f(x) approaches 0 from above or below as x approaches -ve infinity; as x is large the numerator would be +ve but the denominator would be -ve giving f(x) an overall -ve sign. Then f(x) approaches 0 from below.
I find it helpful to draw a sketch graph to check answers.
You might find the Demos graph drawing calculator helpful. The web address is:
https://www.desmos.com/calculator 