Limits The limit of a function at a value is the value that approaches as approaches. Formal Definition

---

Limits 

 

The limit of a function 

at a value  is the value that 

approaches as   approaches .    

 

Formal Definition 

 

  if and only if for every

, there exists   such that 

implies 

. 

 

 

 

Direct Substitution 

 

Some limits can be evaluated by simply substituting   for   in the function 

 

 

Example:  

 

 

Direct substitution works anytime 

 is continuous and 

can be calculated.  This applies to 

constant functions, polynomial functions, and rational functions whenever  is not infinity or 

zero. 

 

However, limits are more often used when

is discontinuous or 

is undefined.   

 

 

 

One-sided Limits 

 

For discontinuous functions (such as some piecewise functions), the limit at a point may not 

exist.  Instead, you can use a right-hand or left-hand limit.  

 

Example: 

 

 

  

Here, the limit at 1 takes on two different values, depending 

on whether you approach 1 from the left (smaller values) or 

from the right (larger values).   

 

 

Right limit:   

 

Left limit: 

 

 

 

When 

, we say that  

 does not exist. 

If  

, then  

 exists. 

 

 

 

---

Limits that approach 

. 

 

Although division by zero is undefined, the limit of a rational function where the numerator 

approaches some positive value and the denominator approaches zero is infinity.  If the 

numerator approaches a negative value, and the denominator approaches zero, the limit is 

negative infinity. 

 

Examples:

                        

 

Often you will be asked to take the limit of a rational function f(x) = p(x)/q(x) where p(x) and 

q(x) both approach zero or infinity.  Here are some tips for finding these limits: 

 

Try to simplify the expression. 

Example:  

  = 

 

 

 

Take the derivative of the top and bottom, and use L’Hopital’s rule: 

 If p(x), q(x) = 0 or 

=

 

Example:    

 

 

Limits at Infinity 

 

Often you will be asked to take the limit of a function as x approaches infinity. 

 

When taking the limit at infinity of a rational function f(x) = p(x)/q(x) where p(x) and q(x) are 

polynomials: 

 

 

If the degree of p is greater than the degree of q, then the limit is positive or negative 

infinity depending on the signs of the leading coefficients;  

 

If the degree of p and q are equal, the limit is the leading coefficient of p divided by the 

leading coefficient of q;  

 

If the degree of p is less than the degree of q, the limit is 0.  

 

Examples:

           

 

 

---

Practice Problems 

 

1.

 

Find each limit using direct substitution. 

a.

 

 

b.

 

 

c.

 

 

 

2.

 

Find  a right-hand and left-hand limit as   approaches   for the function 

 

 

Then determine whether the limit 

exists. 

 

3.

 

 Simplify each expression to find the limit. 

a.

 

 

b.

 

 

 

4.

 

 Find each limit using L’Hopital’s rule. 

a.

 

 

b.

 

 

 

5.

 

Find each limit at infinity using the lead terms of the polynomials. 

a.

 

 

 

b.

 

 

c.

 

 

 

 

 

 

 

Solutions to Practice Problems: