Introduction to Limits

1. Introduction to Limits OBJECTIVE: Calculate a limit

2. Limits of Function Values • Definition of Limit: • If becomes arbitrarily close to a unique number as approaches from either side, the limit of as approaches is . This is written as • Frequently when studying a function, we find ourselves interested in the function’s behavior near a particular point, but not at that point. • Exploring numerically how a function behaves near a particular point at which we cannot directly evaluate because the function leads to division by zero.

3. Finding a limit by using a table of values • Use a table to estimate numerically the limit. • 1. • 2.

4. Finding a limit by using a table of values • 3. • 4. • 5.

5. Finding a limit by using a graph • Using a calculator to guess the limit numerically as x gets closer and closer to c. You discover the behavior of a function near the x-value at which you are trying to evaluate a limit. But sometimes calculators can give false values and misleading impressions for functions that are undefined at a point or fail to have a limit there, because the calculator connects pixels and can’t show the infinitely many oscillations over any interval that contains 0. • 6. 7.

6. Finding a limit by using a graph • 8. • 9. • Windows x-min = -0.25 and x-max = 0.25 and xscl = 0.05, y-min = -1.2 and y-max = 1.2 and yscl = 0.2

7. Conditions under which Limits Do Not Exist • The if any of the following conditions is true. • It jumps: approaches a different number from the right side of c than from the left side of c. • It grows too “large” or too “small” to have a limit: increases or decreases without bound as x approaches c. • It oscillates to much to have a limit: oscillates between two fixed values as x approaches c.

8. Limit Laws • If are real numbers and • 1. Sum Rule: • 2. Difference Rule: • 3. Constant Multiple Rule: • 4. Product Rule: • 5. Quotient Rule: • 6. Power Rule: • 7. Root Rule:

9. EX: Find the following limits • 10. • 12. • 11. • 13.