When learning about the end behavior of a rational function you described the function as either having a horizontal asymptote at zero or another number, or going to infinity. Limit notation is a way of describing this end behavior mathematically.
You already know that as gets extremely large then the function goes to because the greatest powers are equal and is the ratio of the leading coefficients. How is this statement represented using limit notation?
Limit notation is a way of stating an idea that is a little more subtle than simply saying or .
“The limit of of as approaches is ”
The letter can be any number or infinity. The function is any function of . The letter can be any number. If the function goes to infinity, then instead of writing “ ” you should write that the limit does not exist or “ DNE ”. This is because infinity is not a number. If a function goes to infinity then it has no limit.
While a function may never actually reach a height of it will get arbitrarily close to . One way to think about the concept of a limit is to use a physical example. Stand some distance from a wall and then take a big step to get halfway to the wall. Take another step to go halfway to the wall again. If you keep taking steps that take you halfway to the wall then two things will happen. First, you will get extremely close to the wall but never actually reach the wall regardless of how many steps you take. Second, an observer who wishes to describe your situation would notice that the wall acts as a limit to how far you can go.
Translate the following statement into limit notation.
The limit of as approaches 2 is 16.
Translate the following mathematical statement into words.
Solution: The limit of the sum of as the number of terms approaches infinity is 1.
Use limit notation to represent the following mathematical statement.
Concept Problem Revisited
The limit of as approaches infinity is . This can be written using limit notation as:
Limit notation is a way of expressing the fact that the function gets arbitrarily close to a value. In calculus or analysis you may define a limit in terms of the Greek letter epsilon and delta .
1. Describe the end behavior of the following rational function at infinity and negative infinity using limits.
2. Translate the following limit expression into words.
3. What do you notice about the limit expression in # 2?
1. Since the function has equal powers of in the numerator and in the denominator, the end behavior is as goes to both positive and negative infinity.
2. The limit of the ratio of the difference between of quantity plus and of and as approaches 0 is .
3. You should notice that does not mean because if it did then you could not have a 0 in the denominator. You should also note that in the numerator, and are going to be super close together as approaches zero. Calculus will enable you to deal with problems that seem to look like and .
Translate the following statements into limit notation.
6. The limit of as approaches 3 is 19.
7. The limit of as approaches negative infinity is 0.
8. The limit of as approaches infinity is 0.
Use limit notation to represent the following mathematical statements.
10. The series diverges.
Translate the following mathematical statements into words.
15. If , is it possible that ? Explain.