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For all three functions, the limit as x approaches 2 is 7. This brings us to a critical point: When determining the limit of a function as x approaches, say, 2, the value of f 2 — or even whether f 2 exists at all — is totally irrelevant. In a limit problem, x gets closer and closer to the arrow-number c, but technically never gets there, and what happens to the function when x equals the arrow-number c has no effect on the answer to the limit problem though for continuous functions like f x the function value equals the limit answer and it can thus be used to compute the limit answer.
Sidling up to one-sided limits One-sided limits work like regular, two-sided limits except that x approaches the arrow-number c from just the left or just the right. Figure p x : An illustration of two one- sided limits. However, both one-sided limits do exist. As x approaches 3 from the left, p x zeros in on a height of 6, and when x approaches 3 from the right, p x zeros in on a height of 2.
As with regular limits, the value of p 3 has no effect on the answer to either of these one-sided limit problems. And sometimes, like with p x , a piece does not connect with the adjacent piece — this results in a discontinuity. Here goes: Formal definition of limit: Let f be a function and let c be a real number. I think this is why calc texts use the 3-part definition. When we say a limit exists, it means that the limit equals a finite number.
Some limits equal infinity or negative infinity, but you nevertheless say that they do not exist. That may seem strange, but take my word for it. More about infinite limits in the next section. Remember asymptotes? Consider the limit of the function in Figure as x approaches 3. As x approaches 3 from the left, f x goes up to infinity, and as x approaches 3 from the right, f x goes down to negative infinity. But x can also approach infinity or negative infinity.
Limits at infinity exist when a function has a horizontal asymptote. Going right, the function stays below the asymptote and gradually rises up toward it. The following problem, which eventually turns out to be a limit problem, brings you to the threshold of real calculus.
Say you and your calculus-loving cat are hanging out one day and you decide to drop a ball out of your second-story window. After 1. Table Average Speeds from 1 Second to t Seconds As t gets closer and closer to 1 second, the average speeds appear to get closer and closer to 32 feet per second.
It gives you the average speed between 1 second and t seconds: In the line immediately above, recall that t cannot equal 1 because that would result in a zero in the denominator of the original equation. Figure shows the graph of this function.
Chapter 7: Limits and Continuity 85 Figure f t is the average speed between 1 second and t seconds. And why did you get 0? Definition of instantaneous speed: Instantaneous speed is defined as the limit of the average speed as the elapsed time approaches zero.
Linking Limits and Continuity Before I expand on the material on limits from the earlier sections of this chapter, I want to introduce a related idea — continuity. This is such a simple concept. A continuous function is simply a function with no gaps — a function that you can draw without taking your pencil off the paper. Consider the four functions in Figure Whether or not a function is continuous is almost always obvious.
Well, not quite. The two functions with gaps are not continuous everywhere, but because you can draw sections of them without taking your pencil off the paper, you can say that parts of those functions are continuous.
Often, the important issue is whether a function is continuous at a particular x-value. Continuity of polynomial functions: All polynomial functions are continuous everywhere. Continuity of rational functions: All rational functions — a rational function is the quotient of two polynomial functions — are continuous over their entire domains.
They are discontinuous at x-values not in their domains — that is, x-values where the denominator is zero. Consider whether each function is continuous there and whether a limit exists at that x-value. So there you have it. If a function is continuous at an x-value, there must be a regular, two-sided limit for that x-value.
Keep reading for the exception. When you come right down to it, the exception is more important than the rule. Consider the two functions in Figure In each case, the limit equals the height of the hole.
The hole exception: The only way a function can have a regular, two-sided limit where it is not continuous is where the discontinuity is an infinitesimal hole in the function. This bears repeating, even an icon: The limit at a hole: The limit at a hole is the height of the hole. This gave me zero distance. Function holes 0 often come about from the impossibility of dividing zero by zero. The derivative-hole connection: A derivative always involves the unde- fined fraction 0 and always involves the limit of a function with a hole.
Chapter 7: Limits and Continuity 89 Sorting out the mathematical mumbo jumbo of continuity All you need to know to fully understand the idea of continuity is that a func- tion is continuous at some particular x-value if there is no gap there.
You must remember, however, that condition 3 is not satisfied when the left and right sides of the equation are both undefined or nonexistent.
It may seem contrived or silly, but with mnemonic devices, con- trived and silly work. First, note that the word limit has five letters and that there are five 3s in this mnemonic. Remembering that it has three parts helps you remember the parts — trust me. Note that the three types of discontinuity hole, infinite, and jump begin with three consecutive letters of the alphabet.
Hey, was this book worth the price or what? Did you notice that another way this mnemonic works is that it gives you 3 cases where a limit fails to exist, 3 cases where continu- ity fails to exist, and 3 cases where a derivative fails to exist? This chapter gets down to the nitty-gritty and presents several techniques for calculating the answers to limits problems. Easy Limits A few limit problems are very easy. Okay, so are you ready? Limits to memorize You should memorize the following limits.
If you fail to memorize the limits in the last three bullets, you could waste a lot of time trying to figure them out.
The limit is simply the func- tion value. Beware of discontinuities. In that case, if you get a number after plugging in, that number is not the limit; the limit might equal some other number or it might not exist. See Chapter 7 for a description of piecewise functions. What happens when plugging in gives you a non-zero number over zero?
This is the main focus of this section. These are the interesting limit problems, the ones that likely have infinitesimal holes, and the ones that are important for differ- ential calculus — you see more of them in Chapter 9.
Note on calculators and other technology. With every passing year, there are more and more powerful calculators and more and more resources on the Internet that can do calculus for you. A calculator like the TI-Nspire or any other calculator with CAS — Computer Algebra System can actually do that limit problem and all sorts of much more difficult calculus problems and give you the exact answer.
The same is true of websites like Wolfram Alpha www. Many do not allow the use of CAS calculators and com- parable technologies because they basically do all the calculus work for you. Method one The first calculator method is to test the limit function with two numbers: one slightly less than the arrow-number and one slightly more than it. The result, 9. Since the result, The answer is 10 almost certainly.
Method two 2 The second calculator method is to produce a table of values. Hit the Table button to produce the table. Now scroll up until you can see a couple numbers less than 5, and you should see a table of values something like the one in Table These calculator techniques are useful for a number of reasons.
Your calculator can give you the answers to limit problems that are impossible to do alge- braically. Also, for problems that you do solve on paper, you can use your calculator to check your answers.
This gives you a numerical grasp on the problem, which enhances your algebraic understand- ing of it. If you then look at the graph of the function on your calculator, you have a third, graphical or visual way of thinking about the problem.
Many calculus problems can be done algebraically, graphically, and numerically. When possible, use two or three of the approaches. Each approach gives you a different perspective on a problem and enhances your grasp of the relevant concepts. Gnarly functions may stump your calculator. By the way, even when the non-CAS-calculator methods work, these calcu- lators can do some quirky things from time to time.
This can result in answers that get further from the limit answer, even as you input numbers closer and closer to the arrow-number. Try plugging 5 into x — you should always try substitution first. You get 0 — no good, on to plan B. Now substitution will work. And note that the limit as x approaches 5 is 10, which is the height of the hole at 5, Conjugate multiplication — no, this has nothing to do with procreation Try this method for fraction functions that contain square roots.
Try this one: Evaluate lim. Try substitution. Plug in 4: that gives you 0 — time for plan B. Definition of conjugate: The conjugate of a two-term expression is just the same expression with subtraction switched to addition or vice versa. The product of conjugates always equals the first term squared minus the second term squared. Now do the rationalizing. Now substitution works. Plug in 0: That gives you 0 — no good. The best way to understand the sandwich or squeeze method is by looking at a graph.
Look at functions f, g, and h in Figure g is sandwiched between f and h. The limit of both f and h as x approaches 2 is 3. So, 3 has to be the limit of g as well.
Figure The sandwich method for solv- ing a limit. Functions f and h are the bread, and g is the salami. Plug 0 into x. Try the algebraic methods or any other tricks you have up your sleeve. Knock yourself out. Plan C. Try your calculator. It definitely looks like the limit of g is zero as x approaches zero from the left and the right.
Table gives some of the values from the calculator table. Table Table of Values for x g x 0 Error 0. Get it? To do this, make a limit sand- wich. Fooled you — bet you thought Step 3 was the last step. Because the range of the sine function is from negative 1 to positive 1, whenever you multiply a number by the sine of anything, the result either stays the same distance from zero or gets closer to zero.
Thus, will never get above x or below. Figure shows that they do. The function is thus continuous everywhere. Figures and and discussed in the sec- tion about making a limit sandwich. If we now alter it or connect to 0, 0 from the left or the right? The function is now drive through the origin, are you on one of the continuous everywhere; in other words, it has up legs of the road or one of the down legs? But at 0, 0 , it seems to contradict Neither seems possible because no matter how the basic idea of continuity that says you can close you are to the origin, you have an infinite trace the function without taking your pencil number of legs and an infinite number of turns off the paper.
There is no last turn before you reach 0, 0. Now, keeping the line vertical, slowly between you and 0, 0 is infinitely long! It winds up and down with such increas- you pass over 0, 0. There are no gaps in the ing frequency as you get closer and closer to function, so at every instance, the vertical line 0, 0 that the length of your drive is actually infi- crosses the function somewhere.
On this long and function. How can winding road. And because you reconcile all this? I wish I knew. This is confirmed by con- sidering what happens when you plug bigger and bigger numbers into 1 : x the outputs get smaller and smaller and approach zero. Determining the limit of a function as x approaches infinity or negative infinity is the same as finding the height of the horizontal asymptote. That quo- tient gives you the answer to the limit problem and the height of the 3 asymptote.
A horizontal asymptote occurs at this same value. If you have any doubts that the limit equals 0. All you see is a column of 0. Try substitution — always a good idea. No good. On to plan B. Now do the conjugate multiplication. Remember those problems from algebra? One train leaves the station at 3 p. Two hours later another train leaves going east at 50 mph.
You can handle such a problem with algebra because the speeds or rates are unchanging. Think about putting man on the moon. Apollo 11 took off from a moving launch pad the earth is both rotating on its axis and revolving around the sun. On top of all that, the rocket had to hit a moving target, the moon. All of these things were changing, and their rates of change were changing. Much of modern economic theory, for example, relies on differentiation.
In economics, everything is in constant flux. Prices go up and down, supply and demand fluctuate, and inflation is constantly changing. These things are con- stantly changing, and the ways they affect each other are constantly chang- ing. You need calculus for this. Differential calculus is one of the most practical and powerful inventions in the history of mathematics.
The derivative is just a fancy calculus term for a simple idea you know from algebra: slope. Slope, as you know, is the fancy algebra term for steepness. And steepness is the fancy word for. These are big words for a simple idea: Finding the steepness or slope of a line or curve.
Throw some of these terms around to impress your friends. By the way, the root of the words differ- ential and differentiation is difference — I explain the connection at the end of this chapter in the section on the difference quotient.
A steepness of 1 means that as the stickman walks one 2 foot to the right, he goes up 1 foot; where the steepness is 3, he goes up 3 2 feet as he walks 1 foot to the right. This is shown more precisely in Figure Negative slope: To remember that going down to the right or up to the left is a negative slope, picture an uppercase N, as shown in Figure Figure This N line has a Negative slope.
How steep is a flat, horizontal road? Not steep at all, of course. Zero steepness. So, a horizontal line has a slope of zero. Like where the stick man is at the top of the hill in Figure There are more. Variety is not the spice of the wall trying to figure out things like why some 2 author uses one symbol one time and a different mathematics. When mathematicians decide symbol another time, and what exactly does the on a way of expressing an idea, they stick d or f mean anyway, and so on and so on, or 2 to it — except, that is, with calculus.
Hold on to your hat. I all mean exactly the same thing: or df or strongly recommend the second option. I realize that no calculation is necessary here — you go up 2 as you go over 1, so the slope is automatically 2. But bear with me because you need to know what follows. Now, take any two points on the line, say, 1, 5 and 6, 15 , and figure the rise and the run. You rise up 10 from 1, 5 to 6, 15 because 5 plus 10 is 15 or you could say that 15 minus 5 is And you run across 5 from 1, 5 to 6, 15 because 1 plus 5 is 6 or in other words, 6 minus 1 is 5.
Table shows six points on the line and the unchanging slope of 2. That was a joke. So why did I start the chapter with slope? Because slope is in some respects the easier of the two concepts, and slope is the idea you return to again and again in this book and any other calculus textbook as you look at the graphs of dozens and dozens of functions.
A slope is, in a sense, a picture of a rate; the rate comes first, the picture of it comes second. Just like you can have a function before you see its graph, you can have a rate before you see it as a slope. Calculus on the playground Imagine Laurel and Hardy on a teeter-totter — check out Figure Figure Laurel and Hardy — blithely unaware of the calculus implications.
Assuming Hardy weighs twice as much as Laurel, Hardy has to sit twice as close to the center as Laurel for them to balance. And for every inch that Hardy goes down, Laurel goes up two inches. So Laurel moves twice as much as Hardy. A derivative is a rate. You can see that if Hardy goes down 10 inches then dH is 10, and because dL equals 2 times dH, dL is 20 — so Laurel goes up 20 inches. But a rate can be anything per anything.
Again, a derivative just tells you how much one thing changes compared to another. It tells you that for each mile you go the time changes 1 of an hour. We just saw miles per hour and hours per mile. Rates can be constant or changing. In either case, every rate is a derivative, and every derivative is a rate. The rate-slope connection Rates and slopes have a simple connection. All of the previous rate examples can be graphed on an x-y coordinate system, where each rate appears as a slope. Consider the Laurel and Hardy example again.
Laurel moves twice as much as Hardy. The line goes up 2 inches for each inch it goes to the right, and its slope is thus 2 , or 2. One last comment. Well, you can think of dL as the run rise and dH as the run. That ties everything together quite nicely. The Derivative of a Curve The sections so far in this chapter have involved linear functions — straight lines with unchanging slopes.
Calculus is the mathematics of change, so now is a good time to move on to parabolas, curves with changing slopes. You can see from the graph that at the point 2, 1 , the slope is 1; at 4, 4 , the slope is 2; at 6, 9 , the slope is 3, and so on. Unlike the unchanging slope of a line, the slope of a parabola depends on where you are; it depends on the x-coordinate of wherever you are on the parabola. Table shows some points on the parabola 2 and the steepness at those points.
Beginning with the original function, x , take the power and put it in 2 4 front of the coefficient. Reduce the power by 1. In this example, the 2 becomes a 1. So the derivative is 1 x 1 or just 1 x.
The Difference Quotient Sound the trumpets! You come now to what is perhaps the cornerstone of dif- ferential calculus: the difference quotient, the bridge between limits and the derivative.
Okay, so here goes. I keep repeating — have you noticed? You learned how to find the slope of a line in algebra. In Figure , I gave you the slope of the parabola at several points, and then I showed you the short-cut method for finding the derivative — but I left out the important math in the middle. That math involves limits, and it takes us to the threshold of calculus.
For a line, this is easy. You just pick any two points on the line and plug them in. You can see the line drawn tangent to the curve at 2, 4. Figure shows the tangent line again and a secant line intersecting the parabola at 2, 4 and at 10, Definition of secant line: A secant line is a line that intersects a curve at two points. Now add one more point at 6, 36 and draw another secant using that point and 2, 4 again.
Now, imagine what would happen if you grabbed the point at 6, 36 and slid it down the parabola toward 2, 4 , dragging the secant line along with it. Can you see that as the point gets closer and closer to 2, 4 , the secant line gets closer and closer to the tangent line, and that the slope of this secant thus gets closer and closer to the slope of the tangent? So, you can get the slope of the tangent if you take the limit of the slopes of this moving secant.
When the point slides to 2. Sure looks like the slope is headed toward 4. As with all limit problems, the variable in this problem, x2 , approaches but never actually gets to the arrow-number 2 in this case. Herein lies the beauty of the limit process. A fraction is a quotient, right? You may run across other, equivalent ways. Figure is the ultimate figure for. This comprehensive study guide covers every aspect of the exam and the latest revision of the CISSP body of knowledge.
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