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CHAPTER ONE:LIMITS AND PRELUDES TO

CALCULUS

Ch. Desi Kusmindari

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Konstanta = lambang besaran atau kuantitas, 3, 5, -2, 2+5

Paramater = lambang yang mewakili konstanta dalam situasi tertentu,kuntitasnya bervariasi dalam hal-hal berbeda, x = 2t-1 dan y = t2 maka t dalam sistem persamaan tersebut merupakan parameter

Variabel/peubah= lambang kunatitas yang berubah-ubah, 2x-3 = 5, x merupakan variabel

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Perkalian kartesius = perkalian 2 himpunan

Apabila A dan B masing-masing himpunan tidak kosong, maka AxB adalah himpunan pasangan berurutan antara setiap anggota A dengan setiap anggota B

Dirumuskan AxB ={(a,b)|(aA,bB)}Contoh;A = {1,2,3,} dan B = {2,5} maka AxB = {(1,2),(1,5),(2,2),(2,5),(3,2),(3,5)}BxA = {(2,1),….}

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Relasi atau hubungan

Himpunan A dan B dikatakan mempunyai relasi apabila ada cara atau aturan tertentu untuk mengaitkan antara anggota A dengan B.

Dirumuskan ARB atau R : A B={(a,b)|(aA,bB)} A= {1,2,3} dan B {2,5} R1 = lebih kecil, maka R1: AB = {(1,2),(1,5),

(2,5),(3,5)} R2 = sama, maka R2: AB {(2,2)} R3 = lebih besar , maka R3: AB {……}

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Fungsi

Merupakan relasi khusus. Tidak ada pasanngan yang satu anggota domain berpasangan dengan 2 atau lebih anggota kodomain, ditulis f : AB

Jadi fungsi adalah relasi yang memiliki syarat

Contoh: A {2,3} dan B {2,4,9} Fungsi : kuadrat, maka f: AB = {(2,4)}, (3,9)}

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latihan

A={1,2,3} dan B {1,2,4,9}Tentukan dan gambar diagram Vennnya

A x B R1 = AB, bila R1 = lebih kecil R2 = AB, bila R2 = lebih besar R3 = AB, bila R3 = sama F : AB, bila F = kuadrat

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A= {x|x bilangan asli - ganjil}B= {y|y bilangan bulat, -3≤ y <0}Tentukan himpunan semua unsur A x B

yang memenuhi 1 <x <7

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INTRODUCTION

This is a very basic aspect of calculus which needs to be taught first, after reviewing old material.

The concept of limits is very important, since we will need to use limits to make new ideas and formulas in calculus.

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LIMITS

A limit is a mere trend. This trend shows what number y is approaching when x is approaching some number.

Consider this “tilak graph” (parabola) y = x2. Lets get really close with x = 2.

Note that when x is getting really is getting really really close to x=2, then y is getting really really close to 4.

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LIMIT NOTATION

“For the function f(x) = x2, as x gets arbitrarily close to 2, y gets closer and closer to 4.

Or you can use limit notation limx2 x2 = 4

This is how you write limits. If the limit is written in the form limxc f(x) = L, then more than

not, you can substitute the c in to the x. However, limits tell what f(x) value it gets close to while x gets

close to some number, c! It does not necessarily equal that number! In other words:

limxc f(x) does not have to equal f(c)

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LIMITS

Look at this limit. limx5 .

Try to substitute the 5.

Note, that you get 0/0. This is undefined. As you can see, by mere substitution, you cannot always get a good answer.

I claim there is an answer though!

(5)2-25

(5)-5

x2-25

x-5

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LIMITS

You know that at x=5, there should be a point of discontinuity since 5-5=0.

You know that x2 – 25 can be factored to (x+5)(x-5). {{See the algebra needed?}}

Plug those factors into the numerator.

The (x-5)s cancel!

xx22 – 25 – 25 (x+5)(x-5)

x-5x-5

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LIMITS

Just substitute now!limx5 x+5 = L

(5) + 5 = 10 As you can see, simple substitution does not work.

Sometimes you have to use algebra to simplify the expression.

Remember, calculus is the only math that mixes algebra, geometry, and trigonometry together! Factor as much as

possible and cancel enough so you can substitute enough.

If that doesn’t work, then use a graphing calculator to see the trend or look at the graph and see where the

number is getting close to.

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LIMIT PROPERTIES

Very simple things on limits: If c and k are constants and limxc f(x) = L &

limxc g(x) = M ADDITION: limxc f(x) + g(x) = L + M SUBTRACTION: limxc f(x) – g(x) = L – M MULTIPLICATION: limxc f(x) g(x) = (L)(M) CONSTANT: limxc kf(x) =k(L) DIVISION: limxc =

where f(x) and g(x) ≠ 0

f(x)g(x)

LM

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LIMITS INVOLVING INFINITY

There are some limits in which you approach an infinitely big x value and you are looking for an f(x) value that it approaches.

There are also some limits while it approaches a finite numerical x value, you have f(x) values that approach infinity!

Infinity????

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HORIZONTAL ASYMPTOTES

HORIZONTAL: Since we are going infinitely large without

bounds, we must describe x going to infinity, without bound. (∞, “Jagannatha’s eyes”)

lim x±∞ f(x) = 1

In this example, the plus and minus considers the fact that in both +x and –x directions, the

graph is getting closer to y=1.

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HORIZONTAL ASYMPTOTES

Horizontal asymptotes of rational functions are calculated by the following rules. If f(x)= P(x)/Q(x)…

1) See if the degrees of P(x) and Q(x) are equal. If they are, then the horizontal asymptote is found by taking the ratio of the leading coefficients of P(x) and Q(x).

2) If the degree of P(x)> the degree ofQ(x), there is no horizontal asymptote

3) If the degree of P(x) < the degree of Q(x), then the asymptote is at y=0.

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EXAMPLE

f(x)=(2x2+x-1)/(x2-2) Top and bottom degrees are equal (2), so the

ratio of the leading coefficients are 2 to 1 or y=2. f(x)=(2x3+x-1)/(4x2+5x+3) No asymptote since

the top part has a greater degree (3) than the bottom (2).

f(x)= (x2-1)/(x3+3). Since the top degree (2) is smaller than the bottom (3), then the asymptote is y=0.

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VERTICAL ASYMPTOTES

VERTICAL: These type of limits are a little

more complicated. They occur when f(x) values increase very very rapidly without a bound, either positively or negatively. Look at the following example: f(x)=1/(x-3). f(x) is undefined at x=3. Since we are getting very close to x=3 (the undefined point), we are getting very infintely negative from the left and infinitely positive from the right.

NOTE: Unless you can factor and get rid of the denominator, you will have a vertical asymptote as such.

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VERTICAL ASYMPTOTES

The left hand limit notation for the vertical asymptote is as follows.

Left hand side

limx3- f(x) = -∞Right hand side

limx3+ f(x) = ∞

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HOLES AND ASYMPTOTES

This is when you get a hole in the graph. Notice how the original f(x) had x=-2 being undefined. Now, after doing the algebra, you notice that f(x) was just really a line, x-2. But, to take in account that x=-2 being undefined, we have to put a hole in the line, to show ‘inherited’ behavior.

Therefore:

limx-2 f(x)= -4, even though original f(-2) doesn’t equal -4.

2)(

)2(

)2)(2()(

2

4)(

2

xxf

x

xxxf

x

xxf

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END BEHAVIOR

Consider the following rational function.

f(x)= The vertical asymptote is when the denominator equals zero.

Therefore, the vertical asymptote is at x = 4. The graph gets really close to it, but will never touch it. It will get infinitely large without bounds.

The horizontal asymptote is found by taking ratio of the leading coefficients (the coefficients of the greatest exponent) for all degrees of the same power. Since the highest degree is 1, the coefficients are 1:1, therefore, the graph will have a horizontal asymptote at y = 1. The x values would go to infinite numbers but will never touch 1.

The next slide has the graph for f(x) with its two asymptotes.

x-5x-4

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GRAPHICAL VIEW

Notice the two asymptotes. The horizontal on y=1 and the vertical on x=4.

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NUMERICAL METHODS

We have used algebra and graphing techniques to solve for limits.

Sometimes, you have to use numerical methods of obtaining limits. In other

words, for finding limxc f(x) , take a few points really close to c from both sides and evaluate f(x) for point. If both sides approaches a finite number that is equal, then the limit exists.

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EXAMPLE

Using numerical methods, find

limx1 g(x) g(x)= (x-1)/(x³-1) Use the following values of x to help you

find a limit.x= 0.2, 0.4, 0.6, 0.8, 0.9, .99, 1.8, 1.6, 1.4,

1.2, 1.1, 1.01;

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SOLUTION

Obviously, you know that x=1 will be undefined, so you have to see what number g(x) will hover around when x gets really close to 1 on both sides.

Plug in all of those x values in for g(x) and see what trend is formed.

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NUMBER PLUGGING

x g(x)

0.2 0.806452

0.4 0.641026

0.6 0.510204

0.8 0.409836

0.9 0.369004

0.99 0.336689

X g(x)

1.8 0.165563

1.6 0.193798

1.4 0.229358

1.2 0.274725

1.1 0.302115

1.01 0.330022

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LIMIT

Both sides seem to approach x= 1/3. Since g(x) values on both sides of 1 get really

really close to 1/3, the limit is 1/3. The closer you go to x=1 you are getting and

approaching to g(x)=1/3. Consider g(.99) and g(1.01). The numbers are very close to .33333… and it is such that if you were to find g(.9999999.....) or g(1.00000000.....1), then you will get closer to the magic number .333333 repeating (or x=1/3) as the limit.

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CONTINUITY

Another important aspect of functions is to see if it is continuous or not.

A function f(x) is continuous at x=c if it follows all three rules.

1. f(c) must exist.

2. limxc- f(x) = limxc+ f(x)

3. limxc f(x) = f(c)

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ANALYSIS

Is this function continuous at x = 0. 1. f(0) = 0. So, f(0) exists. 2. limits from both sides exists and

are both equal. limx0- f(x) = limx0+ f(x) = 4.

3. However, f(0) and limx0 f(x) = 4 are not equal, therefore, this function is not continuous.

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MY HUMBLE OPINION

In my humble opinion, that is all one must need to know about continuity and limits.

Too much focus on limits will not help us anywhere. Limits here will help us in the future. The introduction of infinity will help us create new formulas and concepts.

Similarly with continuity. Continuity is just included to help define limits and functions in the future. Just remember that limits are a mere trend to see what f(x) value you get to when you infinitely close to a certain x.

Try the following examples.

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END OF CHAPTER PROBLEMS

1. limx2 x3 + 2x + 1

2. limx-6

3. Find all vertical and horizontal asymptotes for the following function. f(x) =

x2-36x+6

x3+3x2-x +36x3-27

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ANSWERS

1. 132. -12

3. limx±∞ f(x) = 1 and

limx3- f(x) = ∞

limx3+ f(x) = -∞

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GENERAL RULES FOR LIMITS

If you are trying to find limxc f(x) . 1) See if f(c) exists or not. This will give you a sense of an idea of

the graph. 2) Check left hand and right hand limits. See what value of f(x) it is

getting close to when x is getting really really close to c. 3) If the right and left hand limits are not equal, overall, there is no

limit 3) If you see a graph increasing rapidly without bound, then the limit

is ∞ depending on its direction. This also means that this limit does not exist.

4) Sometimes, you might have to numerically calculate by having few points less than and greater than c. For example, if c=2, then try having less than c be 1.8, 1.85, 1.9, 1.99, 1.999, etc. and greater than 2 being number like 2.1, 2.01, 2.0001, etc.

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CONCLUSION

Hare Krishna! In conclusion, this chapter was meant to show limits

informally as a definition that will help you out in the next chapters in this course. Calculus is merely algebra, geometry, and trigonometry combined with three new things. 1) limits 2) ∞ 3) the expression 0/0

As for Krsna Conscious examples, this chapter, I admit, didn’t show too much Krsna Conscious examples. Most likely, within the next few chapters, you will see how you can apply calculus to real life Krsna seva. Hare Krsna! Hare Krsna! Krsna Krsna Hare Hare! Hare Rama Hare Rama! Rama Rama Hare Hare!

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CREDITS

This slide show has been re-edited on October 7, 2003 by the help of some people in variety of ways.

Prof. W. Menasco Mr. G. Chomiak Finney Calculus: Graphical, Algebraic, Numerical Mr. J. Trapani A very special thank you to Dr. Hari Surya for reading

the slides, verification, and editing mistakes. Anjali Singh who suggested the numerical method slides

and continuity problems to be added into the slideshow.

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END OF CHAPTER ONE