Limits, Continuity, and Asymptotes

Transcription

1 LimitsContinuity.nb 1 Limits, Continuity, and Asymptotes Limits Limit evaluation is a basic calculus tool that can be used in many different situations. We will develop a combined numerical, graphical, and algebraic approach that will enable us to evaluate a variety of limit forms. In a nutshell, finding the limit of a function as approaches some number, a, means finding what value the function is approaching as (or any independent variable used) approaches the number a. The notation is as follows: lim Øa f() If the function value gets closer and closer to some number, L, as gets closer and closer to a, we say that the lim Øa f() = L. This is sometimes called the general limit. To find this limit always try substituting the number a into the function. If you get a number, that is the limit. Eample 1: Find lim Ø2 2-3 Solution: Substituting 2 for in the function f() = 2-3 gives 2 2-3= 1. So the limit of f() as approaches 2 is 1. Sometimes you run into a problem trying to substitute a number into a function. Eample 2: Find lim Ø2 2-4 ÄÄÄÄÄÄÄÄÄÄÄ -2 Solution: If we try to substitute 2 in for we get ÄÄÄÄÄÄÄÄÄÄ = ÄÄÄÄ This is not good! We know that we cannot divide by zero! Whenever direct substitution fails (especially when we are dealing with a quotient function) we can try other things. In this case, we try factoring the numerator.

2 LimitsContinuity.nb H+2L H-2L ÄÄÄÄÄÄÄÄÄÄÄ = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ -2-2 = +2 So, lim 2-4 Ø2 ÄÄÄÄÄÄÄÄÄÄÄ -2 = lim Ø2+2 = 4. (Note that we can only do this because is getting closer and closer to 2 but never gets there. Otherwise we would be dividing by zero! So if direct substitution fails, try factoring and cancelling and substituting in again. Note that we can get an idea of what the limit of a function is by substituting numbers very close to a in our calculator (this is always a good check if you are allowed to use your calculator). Eample 3: Investigate the limit above numerically using your calculator. Solution: We can do this from our home screen but it is a lot easier if we use our Table function. Enter the function as Y 1. Go to Table Set and set the option to Ask. Now you can enter numbers close to 2 such as 1.9, 1.99, 2.1, and 2.01 and get an idea as to what number, if any, the function is approaching as approaches 2. f(1.9) = 3.9 f(1.99) = 3.99 f(2.1) = 4.1 f(2.01) = 4.01 You can see that as gets closer and closer to 2, f() gets closer and closer to 4. The interesting thing about the eample that we just did is that if you graph the function f() = 2-4 ÅÅÅÅÅÅÅÅÅÅ it looks just like the line f() = +2 (graph it and see!) The only difference -2 is that there is a hole in the graph at the point = 2. The function is not defined at this point. So it seems that even though a function does not eist at a point it can still have a limiting value at that point. We now present an informal definition of the concept of limit (A more precise definition is generally given in more advanced calculus courses but this one will suffice for our use).

3 LimitsContinuity.nb 3 Limit at a point We write lim Øc f() = L or f() ØL as Øc if the functional value f() is close to the single real number L whenever is close to, but not equal to, c (on either side of c). Notice that the definition specifically mentions that the function values approach a single number as approaches c from both sides. Let's investigate this further. Eample 4: Let h() = ÅÅÅÅÅÅ. Eplore the behavior of h() for near 0, but not equal to zero, using a table and a graph. Solution: Note that the function is defined for all ecept 0 (there's that pesky zero in the denominator thing again!) If we make a table of values for h() as approaches 0 from both sides (again use the Table function of your calculator. You get the absolute value signs from the catalog). The results will look like this: Ø h() Ø When is near zero coming from the negative direction h() is -1 and as is near zero coming from the positive direction h() is 1. Consequently we say that lim Øo ÅÅÅÅÅÅ does not eist (or D.N.E) Thus, neither h() nor the limit of h() eist at = 0. However, the limit from the left and the limit from the right both eist at zero, but they are not equal (We will discuss this further below).

4 LimitsContinuity.nb 4 The graph looks like the following: 1 y»» ÅÅÅÅÅÅÅÅÅÅÅ We saw in the above eample that the general limit as approached 0 did not eist but that the function values did approach a single value as we approached zero from the right (1) and another value as we approached from the left (-1). This suggests that the notion of one-sided limits will be very useful when discussing basic limit concepts. One Sided Limits lim Øc -f() = K and call K the limit from the left (or left-hand limit) if f() is close to K whenever is close to c, but to the left of c on the real number line. We write lim Øc +f() = L and call L the limit from the right (or right-hand limit) if f() is close to L whenever is close to c, but to the right of c on the real number line. We now make the following observation: On the Eistence of a Limit In order for a limit to eist, the limit from the left and the limit from the right must both eist and be equal.

5 LimitsContinuity.nb 5 In the last eample, lim Ø0 - ÅÅÅÅÅÅ = -1 and lim Ø0 + ÅÅÅÅÅÅ = 1 Since the left- and right-hand limits are not the same, lim Ø0 ÅÅÅÅÅÅ = does not eist To shorten our work in evaluating limits we can use the following properties: Properties of Limits Let f and g be two functions, and assume that lim Øc f() = L lim Øc g() = M where L and M are real numbers (both limits eist). Then: 1. lim Øc [f() ± g()] = lim Øc f() ± lim Øc g() 2. lim Øc k f() = k lim Øc f() = k L for any constant k. 3. lim Øc [f() g()] = [lim Øc f()] [lim Øc g()] = LM 4. lim Øc f HL ÅÅÅÅÅÅÅÅÅ ghl = lim Øc f H L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ lim Øc gh L = L ÅÅÅÅÅÅ M if M 0 è!!!!!!!!! n 5. lim Øc f HL = è!!!!!!!!!!!!!!!!!!!!! n lim Øc f HL = è!!! n L L>0 for n even Eample 5: If lim Øc f HL = -2and lim Øc ghl= 3, find lim Øc H@ f HLD f HL ghl 2 L

6 LimitsContinuity.nb 6 Solution: Notice that we are trying to find the limit of a sum of products of limits H@ f HLD 2 is after all just f() f()). So we simply "plug in" the individual limits and do the math. H-2L ÿ H-2L ÿ H3L + H3L 2 = = 1 Another concept important to the study of calculus that involves limits is the concept of continuity. Informally, a function is continuous over an interval if its graph can be drawn without removing a pen from the paper. A function that is broken (disconnected) at = c is said to be discontinuous at = c. In general, functions are discontinuous at points where the graph of the function has a hole, a gap, or a vertical asymptote (more about asymptotes later). The formal definition of continuity is as follows: Continuity A function f is continuous at the point = c if 1. lim Øc f() eists 2. f(c) eists 3. lim Øc f() = f(c) A function is continuous on the open interval (a,b) if it is continuous at each point on the interval. If any of the three conditions in the definition fails, then the function is discontinuous at = c. The following properties will enable us to determine intervals of continuity for some important classes of functions without having to look at the graphs or use the three conditions in the definition.

7 LimitsContinuity.nb 7 Continuity Properties of Some Specific Functions A) A polynomial function is continuous for all. Remember that constant, linear, and power functions are special cases of polynomial functions so any of these are continuous for all as well. B) A rational function is continuous for all ecept those values that make a denominator 0. C) The nth root of a function, è!!!!!!!!! n f HL, where n is an odd positive integer greater than 1, is continuous wherever f() is continuous. D) The nth root of a function, è!!!!!!!!! n f HL, where n is an even positive number is continuous wherever f() is continuous and nonnegative. Infinite Limits As we have discovered, a function is discontinuous at any point c where lim Øc f() does not eist. If the left- and right-hand limits do not agree then the limit does not eist. Another situation where a limit might fail to eist involves functions whose values become very large as approaches c. Consider the functions f() = ÅÅÅÅ 1 and g() = ÅÅÅÅÅÅ 1 Notice that as the value of the denominator gets smaller and smaller the function values get larger and larger (try this in your calculator for values of very near zero, both positive and negative). This leads us to the conclusion that lim Ø0 -f() =- and lim Ø0 +f() = Since these two statements represent different types of behavior, we cannot write a single limit statement to describe the nature of the graph at = 0. 2

8 LimitsContinuity.nb 8 The same kind of analysis with g() produces the following results: lim Ø0 -g() = and lim Ø0 +g() = Once again, the limit of g() as approaches zero does not eist, but we can describe the behavior of the graph of g near zero by writing lim Ø0 g() = For both of the functions above, f and g, the line = 0 (the vertical ais) is a vertical asymptote. These ideas are summarized in the following bo.

9 LimitsContinuity.nb 9 Vertical Asymptotes If the limit of a function f fails to eist as approaches c from the left because the values of f() are becoming very large positive (or very large negative numbers) we say that lim Øc -f() = (or - ) If this happens as approaches c from the right, we say that lim Øc +f() = (or - ) If both one-sided limits ehibit the same behavior, we say that lim Øc f() = (or - ) If any of the above hold, the line = c is a vertical asymptote for the graph of y = f(). In summary, if we are dealing with a quotient function then one of two things will happen where the denominator is zero. If the denominator cancels with a factor in the numerator there will be a hole in the graph of the function at that point. If the denominator does not cancel with a factor in the numerator then there will be a vertical asymptote at that point. A little inspection will tell you if the graph is climbing (increasing without bound) or falling (decreasing without bound) as approaches c from either the left or the right. Finally, let's look at limits at infinity. This is known as investigating the end behavior of a function since we are looking at what happens to the graph very far away from the origin. Limits at infinity are written as follows: lim Ø f() Sometimes as the independent variable gets either larger or larger through positive values or larger and larger through negative values so do the function values. In this case we write

10 LimitsContinuity.nb 10 lim Ø f() = ± For instance, if p is a positive real number, then p increase as increases and it can be shown that there is no upper bound on the values of p. We indicate this by writing lim Ø p = Since the reciprocals of very large numbers are very small numbers, it follows that 1 ê p approaches 0 as increases without bound. We indicate this behavior by writing lim Ø 1 ÅÅÅÅÅÅ p = 0 Horizontal Asymptotes If the limit of a function as increases or decreases without bound approaches some number, C, we say that the line y = C is a horizontal asymptote for the graph of the function. There are three very important shortcuts to remember when trying to determine if a function has horizontal asymptotes. All three shortcuts are derived by dividing the numerator and denominator of the quotient function by the highest power of in the numerator and denominator. In actuality all we are doing is multiplying the fraction by 1! Watch the following eample. Eample 6: Find all horizontal asymptotes for the function f() = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Solution: Since the highest power of in the numerator and denominator is 3 we divide each term in the numerator and denominator by 3 resulting in f() = 5 - ÄÄÄÄ 2 ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 4 + ÄÄÄÄÄ 2 ÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄ Now notice that as increases without bound all of the terms with in the denominator will go to zero leaving only the 5 in the numerator and the 4 in the denominator! Therefore, lim Ø f() = ÅÅÅÅ 5 5, so the function has a horizontal asymptote y = ÅÅÅÅ. Use your graphing calculator to investigate the function for large values of 4 4.

11 LimitsContinuity.nb 11 So, one shortcut is to recognize that if the highest power of is the same in the numerator and the denominator the limit as increases without bound is the quotient of the coefficients of the highest power of and.thus, f(0 has a horizontal asymptote at y = this quotient. If the highest power of is in the denominator then the limit as increases without bound is 0 and the function has y = o as a horizontal asymptote. And finally, if the highest power of is in the numerator then the function has no limit and, thus, no horizontal asymptote. That concludes our discussion on limits. Remember that all of the phenomena discussed above can be investigated on your graphing calculator. Worksheet 5a: Limits at a point Worksheet 5b: Eistence of a Limit; Left- and Right-Hand Limits Worksheet 5c: Properties of Limits Worksheet 5d: Continuity Worksheet 5e: Vertical Asymptotes Worksheet 5f: Limits at Infinity and Horizontal Asymptotes