Calculus I, Chapter 3: The Average Rate of Change and Limits
Previous Section: 3.2 The Instantaneous Rate of Change
Limit Notations
In order to solve the problem of instantaneous rates of change equaling zero when we solve them and to describe the process by which we estimate them, we introduce the concept of a limit We denote a limit with the following symbols:
lim┬(x→a)〖f(x)〗=L
In this notation, lim represents the word limit, x→a can be translates as “as x approaches a”, or as x gets infinitely close to a. f(x) is the function whose limit we are evaluating and a is the point at which we are evaluating it. L is the solution to the limit, if it exists. For the sake of this book, we are introducing the limit for the purpose of evaluating the instantaneous rate of change, but we will see it has many more uses later on. The next few sections are dedicated to understanding the limit, how it works, how to solve it and how it is defined.
Solving a Limit
Let’s begin by identifying how to solve a limit. The first method we can take from the previous chapter: using a chart. The chart method for solving a limit involves making an educated guess based on values of the function as the input approaches the specific value. If a function does not approach a specific value, we say that the limit does not exist.
Example 1: Evaluate lim┬(x→3)〖x^2+2x-3〗
Solution:
| x | L |
| 2.9 | 11.21 |
| 2.99 | 11.9201 |
| 2.99 | 11.992001 |
| 3 | 12 |
| 3.001 | 12.008001 |
| 3.01 | 12.0801 |
| 3.1 | 12.81 |
Thus, lim┬(x→3)〖x^2+2x-3〗=12
Example 2: Evaluate lim┬(x→-5)〖3x/(x+5)〗
Solution:
| x | L |
| -5.1 | 153 |
| -5.01 | 1503 |
| -5.001 | 15003 |
| -5 | Does Not Exist |
| -4.999 | -14997 |
| -4.99 | -14997 |
| -4.9 | -147 |
Review
A limit is denoted as lim┬(x→a)〖f(x)〗=L
A limit describes what a functions output approaches as its input tends towards a specific value
To solve a limit we can make a chart that demonstrates what happens when we plug in values close to the specific input value in the limit
It is possible for a limit to not exist, should the chart not approach one single value