lim f ( x) may be given by a table, formula, or a graph. For this handout, we ll assume the function is given by a graph.

Transcription

1 Finding imits Graphicall To find a limit graphicall, we must understand each component of the limit to insure the graph is used properl to evaluate the limit et s look at the smbols used in a limit: lim f ( ) a This limit is called a two sided limit In a tpical two sided limit, ou ll be given the left side of the limit, lim f ( ), and asked to find the right side, A number will be given for a a and ou ll need to find another number The function f ( ) ma be given b a table, formula, or a graph For this handout, we ll assume the function is given b a graph You ma also be asked to evaluate two related limits, lim f ( ) and lim f ( ) a a These limits are called on sided limits To evaluate the two sided limit lim f ( ) a graphicall, we must also evaluate the limits lim f ( ) and lim f ( ) et s eamine what these smbols mean: a a lim f ( ) a As the values get closer to a from values to the left of a on the function f () the values get closer to For a left hand limit, the values are approached from the left side of the number a The value is the value that the graph gets closer to If the values do not get closer to some value, the one sided limit does not eist lim f ( ) a As the values get closer to a from values to the right of a on the function f () the values get closer to For a right hand limit, the values are approached from the right side of the number a The value is the value that the graph gets closer to If the values do not get closer to some value, the one sided limit does not eist

2 If the left hand limit and the right hand limit are both equal to the same number,, then the two side limit is equal to the same number If the one sided limits do not match, the two sided limit does not eist lim f ( ) a As the values get closer to a from values on either side of a on the function f () the values get closer to Eample 1 Find the imit Graphicall Suppose f( ) is given b the graph below a Find lim f( ) 1 To evaluate this limit, we need to eamine values on the graph as gets closer and closer to 1 from the left side of 1 This region of the graph is shown in the graph to the right

3 et s locate an value and its corresponding value in this region Notice that as moves horizontall closer and closer to 1, the corresponding value moves verticall closer and closer to 1 This tells us that lim f( ) 1 1 Notice that the value at 1, f (1) 2, is not the same as the limit b Find lim f( ) 1 In this one sided limit, the values are on the right side of 1 As the point moves to the left towards 1, the point moves up verticall towards 1 This means that the closer the point gets to 1, the closer the value gets to 1 or lim f( ) 1 1 c Find lim f( ) 1 For the two sided limit to eist, the one sided limits must be equal In this case the are both equal to 1 Since the are both equal to 1, the two sided limit is also equal to 1, lim f( ) 1 1 Notice that none of these limits have anthing to do with the fact that f (1) 2 This is due to the fact that we are using values approaching 1, not equal to 1

4 Eample 2 Find the imit Graphicall Suppose f( ) is given b the graph below a Find lim f( ) 1 In the region of the graph slightl to the left of 1, the graph looks eactl like the graph in Eample 1 Notice that as moves horizontall closer and closer to 1, the corresponding value moves verticall closer and closer to 1 This tells us that lim f( ) 1 1 b Find lim f( ) 1 As the point moves to the left towards 1, the point moves up verticall towards 2 This means that the closer the point gets to 1, the closer the value gets to 2 or lim f( ) 2 1

5 c Find lim f ( ) 1 For the two sided limit to eist, the one sided limits must be equal In this case the are not equal From the left side the limit is equal to 1 and from the right side the limit is equal to 2, so lim f( ) does not eist 1 The vertical gap in the graph at 1is what leads to different values in the one sided limits In Eample 1 there was a horizontal gap at 1, but not a vertical gap since the two pieces of the graph come together at 1 Another tpe of limit is a limit at infinit One eample is lim f ( ) As the values get positive and larger and larger on the function f () the values get closer to It is called a limit at infinit because is written as approaching infinit Instead of getting closer and closer to a fied point, the values get larger and larger In this case, we find that the farther to the right we move on the graph, the closer the the values get to the value If the values get ver large (negative of positive) as we move to the right on the graph, then the limit does not eist

6 Eample 3 Find the imit Graphicall Suppose f( ) is given b the graph below a Find lim f( ) To evaluate this limit, we need to eamine values on the graph as gets larger and larger This region of the graph is shown in the graph to the right The horizontal asmptote indicates that the graph gets closer and closer to the horizontal line et s locate an value and its corresponding value Notice that as values grow larger, the corresponding value moves verticall closer and closer to 3 In fact, the more the point moves to the right, the closer it gets verticall to 3 This tells us that lim f( ) 3

7 b Find lim f( ) For a limit where approaches -, we let the values be negative and larger and larger As we move farther and farther to the left on the graph, the corresponding point on the function drops down This means the values are dropping and not approaching a fied value This means the limit does not eist Since the limit does not eist b becoming more and more negative, we write lim f( ) If the values were to become more and more positive because the point rises as we move farther to the left or right, we would similarl conclude that the limit did not eist and then use to indicate how the function values are growing