2.2 Limit of a Function and Limit Laws

Transcription

1 Limit of a Function and Limit Laws Section Notes Page Let s look at the graph y What is y()? That s right, its undefined, but what if we wanted to find the y value the graph is approaching as we get close to an value of? This y-value that it is approaching is called a it Here s some notation for the problem we were just describing What this means is that we want to find what y-value the graph is approaching as gets close to Since we need to get really close to an -value of, let s make a table of values We want to pick values for that are very close to We can pick a couple above and below To find these, you need to put each value into the formula y You can use your table feature on your graphing calculator to do this one Because there are many types of graphing calculators, see me after class if you need to know how to use the table function The table of values should look like: y() By looking at this table, it appears the y-value is approaching 05, or / So you would write your answer as: Now let s use some algebra (this will be eplained more in a later chapter) We can factor the denominator so now the problem becomes To solve this, just put a in for ( )( ) and we get: This is the same as our estimate Limits must approach the same number For eample, let s look at the following graph of f(): Let s look at f ( ) Notice as we approach from the left and from the right we approach two different numbers When this happens, we say the it does not eist

2 What about f ( ) if the graph below is of f()? Section Notes Page Here as approaches from each side, the y values approach two different values, so again the it does not eist What about f ( ) if the graph below is of f()? This one is called an unbounded it since the y-values keep increasing to Since infinity is not a real number, the it does not eist Let s look at the graph below: First, does f() eist? Yes, f() = What is f ( )? Does not eist Approaches two different y-values What is f ( )? This eists As approaches negative one we want to see what y-value the graph approaches This would be zero More on net page

3 Limit Laws Section Notes Page ) f ( ) g( ) f ( ) g( ) c c ) k f ( ) k f ( ) c ) c c c f ( ) g( ) f ( ) g( ) c ) f ( ) c g( ) f ( ) c g( ) c n 5) [ f ( )] f ( ) n c c ) n f ( ) n f ( ) c c 7 5 by using it properties to break it down The properties tell us we can break up this it like this: (5) What are we really doing here? We are simply plugging in the c value into our epression You are not required to show the break down of each it unless the questions specifically ask you to Here, we are just going to replace with since this is our c value () () () 9 The problem with this one is that if I put in a - for I will be dividing by zero which is undefined However if we factor the denominator you will be able to cancel out the part that make the bottom zero Once this part is einated then we can plug in the - for : 9 ( )( )

4 5 8 Section Notes Page This is another one where you will divide by zero if you put in a for Again you would want to factor both the numerator and denominator and then cancel Finally you can then plug in for 5 ( )( ) 8 ( )( ) Since plugging in a won t give us a zero in the denominator, we can just plug in and get the answer: For this one, just plug in - for Remember you are allowed to take the odd root of a negative number 7 0 The problem with this one is that if we put in a zero for we will be dividing by zero so we must do something to this to get rid of the Almost always the operation you will do is to multiply the top and bottom by the conjugate A conjugate (of the numerator in this case) is the same thing but with the opposite sign So we will multiply top and bottom by Then we will cancel: 0 0 Now when we multiply across the top you can use the difference of squares formula, which is ( a b)( a b) a b So if we have then this will equal: and when we simplify we get So let s continue: Now cancel the s to get: Now plug in 0 for : 0 0

5 Section Notes Page 5 This is another one we need to multiply by the conjugate, but this time we will multiply by the conjugate of the denominator, which would be We then will follow the same steps as shown above You can still use the difference of squares formula when you do ( ) Limits with Trigonometric Functions With these its we can still plug in the c value into the epression to get the it You will just need to make sure you have your unit circle or trig tables ready tan Just put in the pi for You will get: tan tan 0 cos cos cos cos 0 sin sin sin Sometimes you may need to use trig identities to simplify before you plug in tan sec If we plug in pi right now we will be dividing by zero We will write these in terms of sine and cosine tan sec sin cos cos sin sin