Limits of Functions

1. Limits of Functions Eric Hoffman Calculus PLHS Sept. 2007

2. Key Topics L is the limit of the function of f as x approaches a, written: if the values of f(x) approach the unique number L as x approaches a from either direction Look at picture on pg. 95 of book

3. Key Topics • Quadratic Function : the limit of a quadratic function is of the form from this we can see that the limit as x approaches a of a quadratic function f is just the value f(a) Ex. Let f(x) = 3x2 – 2x + 3 Functions that have the property are called continuous functions

4. Key Topics • Limit of a function that is not continuous: if a function is not continuous it basically means that the function has an asymptote • Ex. Let this function is undefined at x=3, so if we want to find the limit of this function at x=3 we can’t just plug 3 in for “a”. This is because the function is not continuous at x=3 To solve we must factor out the “offending” factor

5. Key Topics

6. Key Topics • Limits that don’t exist: If we factor the numerator we notice that (x-3) is not a factor, thus we can’t cancel anything out As x approaches 3 the numerator approaches 6 and the denominator approaches 0 thus the quotient “blows up”

7. Key Topics • Properties of limits: let f and g be functions for which and and let c be any real number. Then: Provided m≠0

8. Applying the Properties of Limits • Find

9. Key Topics

10. Key Topics • Homework pg. 100 3-24,multiples of 3 3,6,9… 8 problems!!