Math 220 1.1 Finding Limits Numerically and Graphically( )limx cf x�( )limx cf x� Reads as: The LIMIT of f of x as x approaches “C”☼ When working with limits, we will think of “C” as an “X”, and f(x) is still our normal function notation.( )limcxf x�=☼The “Y” value that is the ANSWER to a limit problem does not have to be f(c). In fact, “f” need not be defined at X=c in order for the limit to exist.Now a few numerical examples:( ) ( )3 2 3 2222 21. Rational Function Find lim lim2 2x xx x x xf x f xx x� �- -= =- -To find the answer to the Limit, we must first compile a table of values for “X” X 1.9 1.99 1.999 2.001 2.01 2.1 F(x) 3.61 3.9601 3.996001 4.004001 4.0401 4.41Numerically the numbers are suggesting that Y is getting closer to 4 as X approaches 2. The Answer to this limit problem is 4. So we say ( )3 22 22lim lim2x xx xf xx� �-=-4=Keep in mind that if 2 was inserted into the function the answer would be INDETERMINATE.( )020f = This means you have a hole in the graph of this function at 2, the X value used. Summary: ( )32222Find lim lim2x xx xf xx� �-=- answer is 4, but the function value at 2, f(2), is 00, or indeterminate.( )3 2222. Piecewise defined function g21 2x xxxxx�-��=-��=�Find ( )3 22 222lim lim21 2x xx xxg xxx� ��-��= =-��=�Once again for this function it is necessary to make a table of values to find the answer to the limit X 1.9 1.99 1.999 2.001 2.01 2.1g(x) 3.61 3.9601 3.996001 4.004001 4.0401 4.41Numerically the numbers are suggesting that Y is getting closer to 4 as X approaches 2.( )3 22 222lim lim21 2x xx xxg xxx� ��-��= =-��=�Note that this function has a function value at x = 2, g(2)=1unlike example one where f(2) was indeterminate.☼ In this example one can see how the function value at x = c and the ( )limx cf x� are not necessarily equal.( )23. h x x= Find ( )22 2lim limx xh x x� �= Yep! Again we need a table of values for X to find our answer. X 1.9 1.99 1.999 2.001 2.01 2.1 h(x) 3.61 3.9601 3.996001 4.004001 4.0401 4.41( )22 2lim limx xh x x� �=4= ( )42h =Graphically: Example 1:( )3 22 22lim lim2x xx xf xx� �-=-( )22x x -=2x -22x x= � -3 -2 -1 1 2 3-2-11234xyGraphically we can see that as X approaches 2 the Y values for the function approach 4, thus the answer to the limit is 4.Example 2:( )( )3 22 2222lim lim41 2 1x xx xxxg xx g x� ��-��-= =��= =�-3 -2 -1 1 2 3-2-11234xyGraphically we can see that as X approaches 2 the Y values for the function approach 4, thus the answer to the limit is 4.Example 3: ( )2h x x= ( )22 2lim limx xh x x� �= -3 -2 -1 1 2 3-2-11234xyThe limit answer for all three examples is 4. Remember the limit answer is not dependent on the function value.☼ There are three basic situations where a limit will not exist.1. When there is a “Break” or “Jump” in the graph, the limit does not exist for that value of C.In this case the X values are not getting close to the same Y thing on both sides.y=f(x)2. When Y values are getting bigger, i.e. closer and closer to infinity, as X gets closer to C. This happens when you have a graph with asymptotes in it.http://www.ltcconline.net/greenl/courses/103a/keys/exam3Practice/exam3PracticeKey.htm3. Oscillation between two fixed numbers as in ( )1sinf xx= also has a limit that DNE. (Does Not Exsit) http://www.wolframalpha.com/The values bunch up at zero, so the limit does not exist.************************************************************************************************************YOU TRY: ( )y f x= =( )1. 2f - = ( )2limxf x�-= ( )2. 1f - = ( )1limxx�-= ( )3. 3f = ( )3limxf x�= ( )4. 4f = ( )4limxf