Limits Video 1 Popper 07 Definition Notation Examples when the domain is all real numbers when there s a hole in the graph piecewise defined functions when there s a vertical asymptote when there s a horizontal asymptote Continuity Video 2 Popper 08 Limits of a sequence Limit of a Difference Quotient Appendix Homework on my website 1 Definition We say that a function f has a limit L as x approaches a when the value for f x can be made as close to L as we want by using x s that are closer and closer but not equal to a Notation lim f x L x a Example 1 Let s look at a nice linear function f x 3x 15 The domain and the range are all Real numbers the y intercept is 15 and the x intercept is 5 Here s the graph Let s look at the limit of f x as x approaches 3 Here s the notation lim 3x 15 L x 3 Now here we have an unusually nice situation to work with The domain and the range are all real numbers On the x axis x can approach 3 from two directions 2 from the left through numbers smaller than 3 x 3 from the right through numbers larger than 3 x 3 and We will make a table of values using some of these numbers and use a calculator to see these values for the function x 3 f x 3x 15 x 3 f x 3x 15 3 1 5 7 3 01 5 97 3 001 5 997 3 0001 5 9997 2 9 6 3 2 99 6 03 2 999 6 003 2 9999 6 0003 The number that we re heading toward is of course 6 Since 6 is the target from above AND below we can say that lim 3x 15 6 x 3 As this is an unusually nice example you may note that the actual value of the function is also 6 f 3 6 lim 3x 15 6 look at x 3 x 3 and what is happening on the Summary x 3 y axis as we approach this number on x on the axes without the graph 3 So taking is limit is a process something is happening on both axes and with the graph points And the limit itself is a number 4 Popper 07 Question 1 5 Example 2 Let s look a another graph f x x 2 4 The domain is all Real numbers The range is 4 The x intercepts are 2 and 2 while the y intercept is 4 Here s the graph f x Let s talk about lim x 2 First let s talk about approaching 2 though numbers less than two coming in from the left x 2 Then we ll talk about approaching 2 through numbers slightly more than 2 coming down the axis from the right x 2 6 As always I ll use a table and calculate the values with a calculator x 2 2 f x x 4 x 2 2 f x x 4 1 9 0 39 1 99 0 0399 1 999 0 003999 1 9999 0 00039999 2 1 0 41 2 01 0 0401 2 0001 0 0004 2 00001 0 00004 As the number we re zooming in on is 0 and we re getting it from both sides we can say lim x 2 4 0 x 2 And since this is a really nice example f 2 0 Let s review our definition from page 1 Definition We say that a function f has a limit L as x approaches a when the value for f x can be made as close to L as we want by using x s that are closer and closer but not equal to a Notation lim f x L x a In example 2 can we get as close as we want to 0 Sure just add some more zeros in the x s that are approaching two and you can get as close to zero as you want 7 Now if everything worked this way we d have no need for limits But it is just not the case that the domain is all Reals all the time Example 3 f x x 2 3x 2 x 2 x 1 x 2 x 1 x 1 This is the line y x 2 with a hole at x 1 Let s look at that graph Note that the point 1 1 is totally missing because the function is given as a rational function This means that 1 is missing from the domain and 1 is missing from the range Domain Range 1 1 1 1 And we want some way to talk sensibly about the graph at the missing point So we take an limit at x 1 x 2 3x 2 x 1 x 1 Find lim 8 x 1 x 2 x 1 x 2 1 1 0 9 1 01 0 99 1 001 0 999 1 0001 0 9999 9 1 1 99 1 01 999 1 001 9999 1 0001 Both rows are heading right for 1 a number that is NOT in the range x 2 3x 2 1 and have some information about the x 1 x 1 We may however say the lim graph The ends of the graph are neatly lined up and are pointing right at one another Note that f 1 is undefined We can have a limit where we do NOT have a graph point 9 Popper 07 Question 2 10 Example 4 Here s a new function f x x x It s NOT a polynomial it IS a sort of rational function but with a twist the absolute value in the numerator Let s look at a few point pairs and then the graph of it First though what s the domain Then x f x 2 2 3 3 10 10 500 500 10 6 10 6 11 Here s the graph f x Let s talk about lim x 0 And we can do it with the usual table and approaching zero from above and below x 0 f x x 0 f x 1 1 01 1 001 1 0001 1 0 1 1 0 01 1 0 001 1 0 0001 1 12 So what we have here is lim f x 1 x 0 and lim f x 1 x 0 two different numbers So we say that this limit does not exist And saying this gives us a different kind of information about this graph The ends are NOT pointing at one another over a one point hole they aren t related at all In symbols lim f x DNE x 0 There s a whole category of functions called Piecewise Defined functions that we ll look at now with an eye toward talking about whether limits at the ends of the pieces exist or not 13 Popper 07 Question 3 14 Piecewise Defined Functions In these functions the domain …