# Introduction to Calculus

## Finding Limits: Properties of Limits

Consider the rational function
The function can be factored as follows:
Does this mean the function 
  is the same as the function 
  
The answer is no. Function 
  does not have 
  in its domain, but 
  does. Graphically, we observe there is a hole in the graph of 
  at 
  as shown in  and no such hole in the graph of 
  as shown in .
So, do these two different functions also have different limits as 
  approaches 7?
Not necessarily. Remember, in determining a limit of a function as 
  approaches 
  what matters is whether the output approaches a real number as we get close to 
  The existence of a limit does not depend on what happens when 
  equals 
  
Look again at  and . Notice that in both graphs, as 
  approaches 7, the output values approach 8. This means
Remember that when determining a limit, the concern is what occurs near 
  not at 
  In this section, we will use a variety of methods, such as rewriting functions by factoring, to evaluate the limit. These methods will give us formal verification for what we formerly accomplished by intuition.

### Finding the Limit of a Sum, a Difference, and a Product

Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits.
Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.

### Finding the Limit of a Polynomial

Not all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the limit of a polynomial function as 
  approaches 
  is equivalent to simply evaluating the function for 
 .

### Finding the Limit of a Power or a Root

When a limit includes a power or a root, we need another property to help us evaluate it. The square of the limit of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots.

### Finding the Limit of a Quotient

Finding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.

### Key Concepts

1. The properties of limits can be used to perform operations on the limits of functions rather than the functions themselves. See .
2. The limit of a polynomial function can be found by finding the sum of the limits of the individual terms. See  and .
3. The limit of a function that has been raised to a power equals the same power of the limit of the function. Another method is direct substitution. See .
4. The limit of the root of a function equals the corresponding root of the limit of the function.
5. One way to find the limit of a function expressed as a quotient is to write the quotient in factored form and simplify. See .
6. Another method of finding the limit of a complex fraction is to find the LCD. See .
7. A limit containing a function containing a root may be evaluated using a conjugate. See .
8. The limits of some functions expressed as quotients can be found by factoring. See .
9. One way to evaluate the limit of a quotient containing absolute values is by using numeric evidence. Setting it up piecewise can also be useful. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, evaluate the limits algebraically.
For the following exercise, use the given information to evaluate the limits: 
  
 .
For the following exercises, evaluate the following limits.
For the following exercises, find the average rate of change 
  

### Graphical

For the following exercises, refer to .

### Real-World Applications
