MATH 1910 - Limits Numerically and Graphically Introduction to Limits The concept of a limit is our doorway to calculus. This lecture will explain what the limit of a function is and how we can find such a limit. Be sure you understand function notation at this point, it will be used throughout the remainder of the course. Consider the function 4 f x x − 16 x−2 Note that the domain of f is x | x ≠ 2. What does the graph look like near x 2? We can certainly graph the function with our graphing calculator & see what happens. Before we do this, though, let’s look at the value (output) of f for values of x close to 2. These can be seen in the following table. x 1.99 1.999 1.9999 2.0001 2.001 2.01 fx ≈ 31.761 31.976 31.998 32.002 32.024 32.241 Note that as x approaches (gets close to) 2, the value of f x seems to be approaching 32. We say 4 "The limit of fx x − 16 as x approaches 2 is 32" x−2 and write 4 lim f x 32 OR lim x − 16 32 x→2 x→2 x − 2 Graph the function over the intervals 0 ≤ x ≤ 4 and 0 ≤ y ≤ 40. Describe what is happening to the graph of f as x approaches 2. Note how the graph shows the same behavior as the table above describes. The following is an intuitive definition for the limit of a function: If f x gets arbitrarily close to a real number L as x approaches (gets close to) a, then f x L OR f x L as x a lim x→a We say this as "the limit of fx as x approaches a is L". The phrase "gets arbitrarily close to" basically means as close as we like. If f x does NOT get arbitrarily close to a real number L, we say that the limit does not exist. I will write DNE from now on if the limit does not exist. Note that in the previous example f 2 does not exist (is undefined), but lim f x DOES exist. x→2 Hence, for a limit to exist at a, the function does not have to be defined at a. Finding Limits Numerically & Graphically When finding limits numerically we will basically construct a table of values as we did in the example above. When finding limits graphically we will look at the graph of the function to estimate limits. Here are some examples: 1. Estimate numerically lim gx if x→9 x −3 x−9 We construct a table of values for gx for values of x close to 9. gx x 8.9 8.99 8.999 9.001 9.01 9.1 gx ≈ 0.16713 0.16671 0.16667 0.16666 0.16662 0.16621 It appears that as x approaches 9 that gx is getting closer to 0.16666... or 0. 1 6 1 6 (see below). 1 Hence, it appears that lim gx 1 x→9 6 Side Note: Do you know how to convert from a non-terminating but repeating decimal expansion like 0. 1 6 to its equivalent fraction? Here’s one way: Let n 0. 1 6 . Then 10n 1. 6 and 100n 16. 6 . Thus 90n 100n − 10n 16. 6 − 1. 6 15 Thus 90n 15 n 15 1 90 6 sin x 2. Construct a table of values for f x x for x close to zero to estimate x lim sin x x→0 What mode should we be in? (radian or degree?) Here is such a table: x 0. 1 0. 01 0. 001 sin x ≈ 0.99833417 0.9999833 0.9999998 x Thus it appears that x lim sin x 1 x→0 This is an important limit we will see again. Look at the graph of sinx x over the intervals − ≤ x ≤ & −2 ≤ y ≤ 2 to confirm the numerical approach. 3. An example of a limit that does not exist (DNE). Consider the function f x sin 1x . Note that the domain of f is all real numbers except 0. What can we say about lim sin 1x x→0 When we try to graph this function for values of x near zero, our graphing calculator has problems. Graph f over the intervals −2 ≤ y ≤ 2 and (1) −3 ≤ x ≤ 3, (2) −1 ≤ x ≤ 1, (3) −0. 1 ≤ x ≤ 0. 1, and finally (4) −0. 01 ≤ x ≤ 0. 01. What do you observe? What is happening? Recall that 1 4n sin t 1 if t 2n 4n for n 0, 1, 2, 3, . . . 2 2 2 2 and 3 4n for n 0, 1, 2, 3, . . . sin t −1 if t 3 2n 3 4n 2 2 2 2 So 1 4n 2 or when x for n 0, 1, 2, 3, . . . sin 1x 1 when 1x 2 1 4n Thus f x 1 when 2 , 2 , 2 , 2 ,..., 2 x ,... 5 9 13 1 4n Note that as n → , x → 0. Or saying it another way, between any positive number x and zero, there are an INFINITE number of times when f x 1. In a similar manner, 2 3 4n 2 sin 1x −1 when 1x or when x for n 0, 1, 2, 3, . . . 2 3 4n Thus f x −1 when 2 x 2 , 2 , 2 , 2 ,..., ,... 3 7 11 15 3 4n Note that as n → , x → 0. Or saying it another way, between any positive number x and zero, there are an INFINITE number of times when f x −1. Thus we see that as x gets close to zero, f x begins to wildly oscillate between −1 and 1. In essence, f x can never "settle down" and approach any one limit. Thus lim sin 1x DNE. x→0 4. When Technology Fails. We saw in the last example that our graphing calculator had troubles graphing the function for values of x close to zero. This example shows another type error we can run into. t4 1 − 1 If gt , estimate numerically the following limit t4 lim gt t→0 We proceed by constructing a table of values for x close to zero t 0. 1 0. 01 0. 001 0. 0001 gt 0.49999 0.5 0 0 Up to x 0. 01, we may guess that the limit appears to be approaching 12 . But, if we get closer to zero, we see the limit appears to be zero. What is going on? The problem is that when your graphing calculator evaluates t 4 1 for small values of t, the result is very close to 1. In fact, if you plug in t 0. 001 in the TI-84 the result is zero. This is due to the limitations of the graphing calculator & the number of digits the calculator is able to carry (Graph g with 0 ≤ y ≤ 1 over (1) −4 ≤ x ≤ 4 and the (2) −0. 01 ≤ x ≤ 0. 01). The value of this limit is 12 which will be able to show later. One-Sided Limits Consider the function f x 1 if x 2 3 if x 2 The graph of f is shown below 4 3 2 1 -2 2 4 6 Note that as x approaches 2 from the left (or from the negative side or from below) fx approaches 1 (it is always 1 for x 2). But as x approaches 2 from the right (or from the positive side or from above) fx approaches 3. Since we do not approach any ONE value from both "sides", lim fx DNE. x→2 When fx approaches 1 as x approaches 2 from the left we write 3 lim f x 1 x → 2− and we say "the limit of fx as x approaches 2 from the left is 1" or "the left-hand limit of fx as x approaches 2 is 1". Similarly, When fx approaches 3 as x approaches 2 from the right we write lim f x 3 x → 2 and we say "the limit of fx as x approaches 2 from the right is 3" or "the right-hand limit of fx as x approaches 2 is 3". With one-sided limits we have the following useful theorem lim f x L if and only if xlim f x L xlim f x x→a →a− →a fx L if and only if both one-sided limits exist and are both equal to L. That is, lim x→a Consider the function −1 if x 0 |x| x 1 if x 0 The graph of the function is shown below 1 -4 -2 2 4 -1 |x| |x| Note that lim x −1, but lim x 1. Both one-sided limits exist, but they are not equal. Thus x→0− x→0 |x| lim x DNE. x→0 Consider the function hx whose graph is shown below. Find the following limits (if they exist) (a) lim hx (b) lim hx (c) lim hx (d) lim hx (e) lim hx (f) lim hx x → −2 − x → −2 x → −2 x→1− x→1 x→1 4 2 -5 -4 -3 -2 -1 1 2 3 4 5 -2 -4 Infinite Limits Let’s try to find the limit lim 12 We proceed numerically, constructing a table of values for x close to x→0 x zero. x 1 x2 0. 1 0. 01 0. 001 0. 0001 100 10,000 1,000,000 100,000,000 4 As x gets closer to zero, the value of 12 continues to get bigger and bigger. It does NOT approach any x finite number. Thus, we approach no FINITE limit. To indicate this kind of behavior we introduce the notation lim 12 x→0 x and say that f has an infinite limit. Note that this does NOT mean is a number (which it is not). It simply expresses the idea that the value of the function gets arbitrarily large (as large as we want) as x gets close to 0. When we see this expression, we say "the limit is infinity" or "the function increases without bound". If a function f gets arbitrarily large BUT NEGATIVE as x approaches a, we write lim fx − x→a We can say similar statements with one-sided limits. Examples: 1. For the function g shown below find the following limits or write DNE. lim gx lim gx lim gx x → −2 − x → −2 lim gx x → −2 lim gx x→3− lim gx x→3 x→3 10 5 -4 -3 -2 -1 1 2 3 4 -5 -10 2. Show that lim ln x −. x→0 5

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