# Introduction to Calculus

## Finding Limits: Numerical and Graphical Approaches

Intuitively, we know what a limit is. A car can go only so fast and no faster. A trash can might hold 33 gallons and no more. It is natural for measured amounts to have limits. What, for instance, is the limit to the height of a woman? The tallest woman on record was Jinlian Zeng from China, who was 8 ft 1 in.https://en.wikipedia.org/wiki/Human_height and http://en.wikipedia.org/wiki/List_of_tallest_people Is this the limit of the height to which women can grow? Perhaps not, but there is likely a limit that we might describe in inches if we were able to determine what it was.
To put it mathematically, the function whose input is a woman and whose output is a measured height in inches has a limit. In this section, we will examine numerical and graphical approaches to identifying limits.

### Understanding Limit Notation

We have seen how a sequence can have a limit, a value that the sequence of terms moves toward as the nu mber of terms increases. For example, the terms of the sequence
gets closer and closer to 0. A sequence is one type of function, but functions that are not sequences can also have limits. We can describe the behavior of the function as the input values get close to a specific value. If the limit of a function 
  then as the input 
  gets closer and closer to 
  the output y-coordinate gets closer and closer to 
  We say that the output “approaches” 
  
 provides a visual representation of the mathematical concept of limit. As the input value 
  approaches 
  the output value 
  approaches 
  
We write the equation of a limit as
This notation indicates that as 
  approaches 
  both from the left of 
  and the right of 
  the output value approaches 
  
Consider the function
We can factor the function as shown.
Notice that 
  cannot be 7, or we would be dividing by 0, so 7 is not in the domain of the original function. In order to avoid changing the function when we simplify, we set the same condition, 
  for the simplified function. We can represent the function graphically as shown in .
What happens at 
  is completely different from what happens at points close to 
  on either side. The notation
indicates that as the input 
  approaches 7 from either the left or the right, the output approaches 8. The output can get as close to 8 as we like if the input is sufficiently near 7. 
What happens at 
  When 
  there is no corresponding output. We write this as
This notation indicates that 7 is not in the domain of the function. We had already indicated this when we wrote the function as
Notice that the limit of a function can exist even when 
  is not defined at 
  Much of our subsequent work will be determining limits of functions as 
  nears 
  even though the output at 
  does not exist.

### Understanding Left-Hand Limits and Right-Hand Limits

We can approach the input of a function from either side of a value—from the left or the right.  shows the values of
as described earlier and depicted in .
Values described as “from the left” are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left in  are 
  
  and 
  The corresponding outputs are 
  and 
  These values are getting closer to 8. The limit of values of 
  as 
  approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function 
  as 
  approaches 7.
Values described as “from the right” are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right in  are 
  
  and 
  The corresponding outputs are 
  
  and 
  These values are getting closer to 8. The limit of values of 
  as 
  approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function 
  as 
  approaches 7.
 shows that we can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input 
  within the interval 
  to produce an output value of 
  within the interval 
  
We also see that we can get output values of 
  successively closer to 8 by selecting input values closer to 7. In fact, we can obtain output values within any specified interval if we choose appropriate input values.
 provides a visual representation of the left- and right-hand limits of the function. From the graph of 
  we observe the output can get infinitesimally close to 
  as 
  approaches 7 from the left and as 
  approaches 7 from the right.
To indicate the left-hand limit, we write
To indicate the right-hand limit, we write

### Understanding Two-Sided Limits

In the previous example, the left-hand limit and right-hand limit as 
  approaches 
  are equal. If the left- and right-hand limits are equal, we say that the function 
  has a two-sided limit as 
  approaches 
  More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.

### Finding a Limit Using a Graph

To visually determine if a limit exists as 
  approaches 
  we observe the graph of the function when 
  is very near to 
  In  we observe the behavior of the graph on both sides of 
  
To determine if a left-hand limit exists, we observe the branch of the graph to the left of 
  but near 
  This is where 
  We see that the outputs are getting close to some real number 
  so there is a left-hand limit.
To determine if a right-hand limit exists, observe the branch of the graph to the right of 
  but near 
  This is where 
  We see that the outputs are getting close to some real number 
  so there is a right-hand limit.
If the left-hand limit and the right-hand limit are the same, as they are in , then we know that the function has a two-sided limit. Normally, when we refer to a “limit,” we mean a two-sided limit, unless we call it a one-sided limit.
Finally, we can look for an output value for the function 
  when the input value 
  is equal to 
  The coordinate pair of the point would be 
  If such a point exists, then 
  has a value. If the point does not exist, as in , then we say that 
  does not exist.

### Finding a Limit Using a Table

Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of 
  approach 
  from both sides. Then we determine if the output values get closer and closer to some real value, the limit 
  
Let’s consider an example using the following function:
To create the table, we evaluate the function at values close to 
  We use some input values less than 5 and some values greater than 5 as in . The table values show that when 
  but nearing 5, the corresponding output gets close to 75. When 
  but nearing 5, the corresponding output also gets close to 75.
Because
then
Remember that 
  does not exist.

### Key Concepts

1. A function has a limit if the output values approach some value 
  as the input values approach some quantity 
  See .
2. A shorthand notation is used to describe the limit of a function according to the form 
  which indicates that as 
  approaches 
  both from the left of 
  and the right of 
  the output value gets close to 
  
3. A function has a left-hand limit if 
  approaches 
  as 
  approaches 
  where 
  A function has a right-hand limit if 
  approaches 
  as 
  approaches 
  where 
  
4. A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit.
5. A graph provides a visual method of determining the limit of a function.
6. If the function has a limit as 
  approaches 
  the branches of the graph will approach the same 
  coordinate near 
  from the left and the right. See .
7. A table can be used to determine if a function has a limit. The table should show input values that approach 
  from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit. See .
8. A graphing utility can also be used to find a limit. See .

### Section Exercises


### Verbal


### Graphical

For the following exercises, estimate the functional values and the limits from the graph of the function 
  provided in .
For the following exercises, draw the graph of a function from the functional values and limits provided.
For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as 
  approaches 0.
For the following exercises, use a graphing utility to find graphical evidence to determine the left- and right-hand limits of the function given as 
  approaches 
  If the function has a limit as 
  approaches 
  state it. If not, discuss why there is no limit.

### Numeric

For the following exercises, use numerical evidence to determine whether the limit exists at 
  If not, describe the behavior of the graph of the function near 
  Round answers to two decimal places.
For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as 
  approaches the given value.
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as 
  approaches 
  If the function has a limit as 
  approaches 
  state it. If not, discuss why there is no limit.

### Extensions
