1.5 LIMITS Calculus 9/16/14. WARM-UP LIMITS – P. 49.

1.5 LIMITSCalculus 9/16/14

WARM-UP

• 1. What is the domain and range:

2. Evaluate the expression and simplify ,

LIMITS – P. 49

GOAL

•Find limits graphically or numerically•Use properties of limits and analytic techniques to evaluate limits of a function

DEFINITION

•Limit of a Function – If becomes arbitrarily close to a single number as approaches from either side, then

“the limit of as approaches c is ”

FINDING A LIMIT•Direct Substitution:

•

EX. 1

•

a) c) graph

b) Table

EXAMPLE 2.•

•

• table

•Graph

EXAMPLE 3

• table

•Graph

EX. 3

• Graph

FIND THE LIMIT

lim𝑥→ 1

𝑥3+4 𝑥−5𝑥−1

PROPERTIES –P.51

OPERATIONS

EX 6- FIND THE LIMIT

lim𝑥→ 2

𝑥2+2 𝑥−3

WARM-UP•Graph

•What does the graph look like and why?

•When does a limit not exist?

LIMTS•Limits are useful when we need to find the value of a function getting close to a certain “x” value•We can use direct substitution (“plug in” the x to find the value at that point), but sometimes that is not possible and limits help us see what the function value could be as we approach that x value

FIND THE LIMIT

•

FIND THE LIMIT

lim𝑥→ 1

𝑥3−1𝑥−1

ONE- SIDED LIMITS

•

lim𝑥→ 0+¿|2 𝑥|

𝑥¿

¿

DOES THE LIMIT EXIST OR NOT?

FIND A ONE-SIDED LIMIT

• Find the limit of f(x) as x approaches 1

UNBOUNDED BEHAVIOR

lim𝑥→ 2

3𝑥−2

COMPARING ONE SIDED LIMITS

•Ex. 9 in the book.