Calculus

Introduction to Limits

Limit notation is a way of stating an idea that is a little more subtle than simply saying x=5 or y=3.

The letter a can be any number or infinity. The function f(x) is any function of x. The letter b 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.

Take the following limit:

The limit of y=4x as x approaches 2 is 16

In limit notation, this would be:

While a function may never actually reach a height of b it will get arbitrarily close to b. 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.

You should notice that h→0 does not mean h=0 because if it did then you could not have a 0 in the denominator. You should also note that in the numerator, f(x+h) and f(x) are going to be super close together as h approaches zero. Calculus will enable you to deal with problems that seem to look like / and / .

End behavior

End behavior is a description of the trend of a function as input values become very large or very small, represented as the ‘ends’ of a graphed function.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that indicates where a function flattens out as the independent variable gets very large or very small. A function may touch or pass through a horizontal asymptote.

limit notation

Limit notation is a way of expressing the fact that a function gets arbitrarily close to a value.