1. Limits of Functions

2. What do we mean by limit? In everyday conversation, we use the word “limit” to describe the ultimate behavior of something. e.g. “the limit of one’s patience”

3. Limit of a Function  Is the number that f(x) approaches as x approaches c.  So…we are only concerned about what happens with f(x) as x gets arbitrarily close to c.  We don’t actually care about what happens when x=c.

4. How do we find limits? There are several methods, but we will use the TABLE METHOD. So let’s use this method…

5. Our First Example  The left column of our table will contain numbers approaching 3.  The right column of our table will contain the value of these numbers evaluated in the function 2x+4

6. How do we approach 3? We have two choices: 1) Approach 3 from the left 2) Approach 3 from the right Approaching from the left will give the Left-Sided Limit Approaching from the right will give the Right-Sided Limit

7. The Left-Sided Limit x2x+4 2.9 2.99 2.999 2.9999 2.99999 9.8 9.98 9.998 9.9998 9.99998 The column on the left contains the values we chose that are LESS than 3 and are approaching 3. The values in the right column are approaching 10. Thus, 10 is the Left Sided Limit!

8. The Right-Sided Limit x2x+4 3.1 3.01 3.001 3.0001 3.00001 10.2 10.02 10.002 10.0002 10.00002 The column on the left contains the values we chose that are GREATER than 3 and are approaching 3. The values in the right column are approaching 10. Thus, 10 is the Right- Sided Limit!

9. Are we done? Almost… We were simply asked to find the limit of the function, not the left or right sided limits. However, since the left and right sided limits both were equal to 3, we say that the limit of the function is 3.

10. One more thing… If the Left-Sided Limit and Right- Sided Limit of a function do not agree, then we say that the limit of the function DOES NOT EXIST. In conclusion, remember to always…