Chapter 3 Notes-1184556490

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3.1 - Limits Math 130 - Price Main Idea: A limit describes the behavior of a function (y-value), as the value of x gets closer and closer to a specific value (but not necessarily at that value) For any number "a" and a function ( ) f x Limit from the left: The limit as x approaches a from the left is written Limit from the right: The limit as x approaches a from the right is written Two-sided limit: The limit as x approaches a is written LIMITS FROM A GRAPH Ex 1: Use the graph to find the following: a) ( ) 2 lim x f x b) ( ) 2 lim x f x + c) ( ) 2 f d) ( ) 2 lim x f x *TRY IT* Ex 2: Use the graph to find the following: a) ( ) 1 lim x f x b) ( ) 1 lim x f x + c) ( ) 1 f d) ( ) 1 lim x f x
A limit is a tool that is used to describe the behavior of a function as the value of x approaches some particular number. (not necessarily AT that number) Limit: Let f be a function and let a and L be real numbers. For a function f ( ) lim x a f x L = means that as x approaches a (from the left and right side), ( ) f x approaches L Written in terms of limits, we can say: If ( ) lim x a f x L = AND ( ) lim x a f x L + = , then ( ) lim x a f x L = *Notes about limits (this addresses common misunderstandings) : 1) If the left hand limit does not equal the right hand limit, then ______________________________ __________________________________________________________ 2) The existence of the limit ( ) lim x a f x does NOT depend on whether ________________________ 3) The existence of the limit ( ) lim x a f x does NOT depend on the value of ____________________
*TRY IT* Ex 3: If ( ) 7 lim 4 = x f x , ( ) 7 lim 4 = + x f x and ( ) 2 4 = f , find ( ) 4 lim x f x Ex 4: If ( ) 8 lim 2 = x f x and ( ) 10 lim 2 = + x f x , find ( ) 2 lim x f x LIMITS USING A TABLE : Ex 5: Find 2 2 4 lim 2 x x x by using a table of values. First, notice what happens when we plug in 2 x = into 2 4 2 x x We want to examine what happens to the function 2 4 ( ) 2 x f x x = as values of x get closer and closer to 2 , from the left AND right side. x 1.9 1.99 1.999 2 2.001 2.01 2.1 ( ) f x
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