Pre-Calculus Notes Section 12.2 Evaluating Limits DAY ONE: Lets look at finding the following limits using the calculator and algebraically.

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1 Pre-Calculus Notes Name Section. Evaluating Limits DAY ONE: Lets look at finding the following its using the calculator and algebraicall. 4 E. ) 4 QUESTION: As the values get closer to 4, what are the values getting closer to? NOTATION: 4 4 or as, Could we simpl substitute the 4 in for in So how do we do this problem without a calculator? 4? Wh NOT? ( )( ) ( 4) Now what do we do to get the value to goes with 4? Remember: We call this the CANCELLATION METHOD OR DIVIDING OUT TECHNIQUE E. ) + WITH CALCULATOR Go to table and look at the values as the approach. What are the values approaching? Do ou see a value for when? Wh not? Put our calculator on the ASK table function and complete the following table Could we simpl substitute the in for in f ( )? + Wh NOT? So how do we do this problem without a calculator?

2 f ( ) + ( ) ( )( ) Now what do we do to get the value to goes with? We call this the new function we found a function that agrees with the original function at all but one point. Find the following its b using the dividing out technique and finding a function that agrees with the original in all but one point E. ) E. 4) 5 5 E. 5) 4 + E. 6) Now, let s mi some substitution and dividing out. A) Graph the function and determine the its visuall (if the eist) B) Find another function that agrees with the given function at all but one point. + E. 7) f ( ) (B) Now, let s practice using our calculator AGAIN to get its. E. 8) 0 cos() ( + ) E. 9) 0 ( / )

3 Let s look AGAIN at finding the following its using the calculator and algebraicall + E. 0) 0 WITH CALCULATOR Go to table and look at the values as the approach 0. What are the values approaching? Do ou see a value for when 0? Wh not? Put our calculator on the ASK table function and complete the following table 0 + Could we simpl substitute the 0 in for in +? Wh NOT? So how do we do this problem without a calculator? We will use a technique call RATIONALIZING THE NUMERATOR we multipl b the conjugate of the numerator!!!! + + ( ) ( ) Hint: DO NOT MULTIPLY THE DENOMINATIORS OUT! (We want to cancel the out.) Now what do we do to get the value to goes with 0? E. ) 0 z z z 7

4 E. ) z 4 8 z z DAY TWO: Now, let s look at some properties of its. Eample ) Given: f ( ) and g ( ) A) 5( ) 5( ) What do ou notice? B) ( + ) What do ou notice? C) ( )( ) What do ou notice? PROPERTIES OF LIMIITS Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following its: f ( ) L g( ) Scalar Multiple: c Product: c Power: c ( bf ( )) bl and K c ( f ( ) g( )) LK ( n f ( )) L n c Sum or Difference: c Quotient: ( f ( ) ± g( )) L ± K f ( ) g( c ) L K

5 E ) Using the above properties, find the following its. f ( ) 7 c and g( ) c 0 ( ) a. ( f ) b. ( f( ) g ( )) c c 5 ( ) ( ) c. ( f g) c d. f( ) g ( ) c e. ( ( )) f c g ( ) f. ( ) 5 c Let s look AGAIN at finding the following its using the calculator and algebraicall. 4 4 E. ) 0 WITH CALCULATOR: Put our calculator on the ASK table function and complete the table Could we simpl substitute the 0 in for in f ( ) 4 4? Wh NOT? So how do we do this problem without a calculator? We will use a technique called SIMPLIFYING THE COMPLEX FRACTION. Hint: Multipl the top and bottom b the LEAST COMMON DENOMINATOR. f ( ) 4 4

6 Now what do we do to get the value to goes with 0? E. 4) Now, let s look at left- hand and right- hand its and functions that do not have its at particular -values. + < f( ) 4 E. 5) A) C) 0 0 f( ) f( ) B) D) 0 + f( ) f( ) E) + f( ) F) f( ) E. 7) A) π cos 0 0 f( ) B) 0 + f( ) C) 0 f( ) Hint: This is an unusual graph. Pick intervals closer and closer to 0 for eample and, then -. and., then -.0 and.0 and so on. What seems to be happening?

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