LIMIT

CONTOH LIMIT FUNGSI ALJABAR

1. l i m x² - 5x + 6 = (3)² - 5(3) + 6 = 0
x ® 3

2. l i m 3x - 2 = ¥ (*) Uraikan
x ® ¥ 2x + 1 ¥

x(3 - 2/x) = 3 - 2/x = 3 - 0 = 3
x(2 - 1/x) 2 + 1/x 2 - 0 2

atau langsung gunakan hal khusus

3. l i m x² - x - 1 = ¥ (*) Uraikan
x ® ¥ 10x + 9 ¥

x(x - 1 - 1/x) = x - 1 - 1/x = ¥ - 1 - 0 = ¥ =¥
x(10 - 9/x) 10 + 9/x 10 + 0 10

atau langsung gunakan hal khusus


4. l i m x² - 3x + 2 = 0 (*) Uraikan
x ® 2 x² - 5x + 6 0

(x - 1)(x - 2) = (x - 1) = 2 - 1 = -1
(x - 3)(x - 2) = (x - 3) = 2 - 3

atau langsung gunakan hal khusus ® Differensial


5. l i m x³ - 3x² + 3x - 1 = 0 (*) Uraikan
x ® 1 x² - 5x + 6 0

(x - 1)³ = (x - 1)² = (1 - 1)² = 0
(x - 1) (x - 5) (x + 5) (1 + 5) 6

atau langsung gunakan hal khusus ® Differensial



6. l i m Ö² + x - Ö2x = 0 (*) Hilangkan tanda akar dengan
x ® 2 x - 2 0 mengalikan bentuk sekawan

(x - 1)³ = (x - 1)² = (1 - 1)² = 0 = 0
(x - 1) (x - 5) (x + 5) (1 + 5) 6

atau langsung gunakan hal khusus ® Differensial



7. l i m (^3x - Ö9x² + 4x) = ¥ - ¥ (*) Hilangkan tanda akar
x ® ¥

l i m (^3x - Ö9x² + 4x ) = é ^3x - Ö9x² + 4x ù = (*) Hilangkan tanda
x ® ¥ ë ^3x - Ö9x² + 4x û akar

l i m (^9x2 - (9x² + 4x) = l i m -4x =
x ® ¥ 3x + Ö(9x² + 4x) x ® ¥ 3x + 3x Ö[1+(a/9x)]

l i m -4 = -4 = -2
x ® ¥ 3 + 3Ö⁽¹ + 0) 6 3

atau langsung gunakan hal khusus

CONTOH LIMIT FUNGSI TRIGONOMETRI

1. l i m sin 2x = 0 (*)
x ® 0 tg 3x 0

sin 2x = 3x 2 = 1 . 1 . 2 = 2
2x tg 3x 3 3 3

2. l i m 1 - cos 2x = 0
x ® 0 sin 2x 0

1 - (1 - 2 sin² 2x) = 2 sin² x = sin x = tg x = 0
2 sin x cos x 2 sin x cos cos x

3. l i m 1 - cos x = 0
x ® 0 3x² 0

2 sin² (½x) = sin (½x) . sin (½x) = 1 . 1 . 1 = 1
3 . 4 . (½x) 6 (½x) (½x) 6 6

atau langsung gunakan hal khusus ® Differensial

trigonometry

The word "Trigonometry" is derived from two Greek words meaning measurement or solution of triangles. Trigonometry is a branch of mathematics that deals with the ratio between the sides of a right triangle and its angles. Trigonometry is used in surveying to determine heights and distances, in navigation to determine location and distances, and in fields like nondestructive testing for determining things such as the angle for reflection or refraction of an ultrasound wave.

There are three principle functions in trigonometry: sine A, cosine A, and tangent A where A is an angle. These are typically abbreviated for use in algebra to: sinA, cosA, and tanA. These terms are defined in terms of a right triangle. SinA is equal to the side opposite of the angle A (side a) divided by the hypotenuse of the triangle (side c). From the right triangle below, it can be seen that the value of angle A is directly linked to the ratio of side a and side c. In other words, if the length of side a is changed (and side c is not changed by the same amount), then angle A must change. When the division of the length of side a by side c is performed, the resulting value is directly related to angle A. The relationship between the decimal value of the ratio of the side and the angular value of angle A can be looked up in trigonometry tables or, as is more common these days, programmed into a scientific calculator. The relationship between angle A and the ratio of any two of the three sides can be determined by using either the Sin, Cos or Tan functions. CosA is equal to the side adjacent to the angle A divided by the hypotenuse and tanA is the sine divided by the cosine and is therefore the side opposite the angle divided by the side adjacent.

Angles can be measured in either radians or degrees. 180º = p radians.

Trigonomic Properties

There are several rules to make trigonometry easier. The first rule is the law of sines. This rule is valid for all triangles and is not restricted to right triangles. The law of sines is shown below.

The second rule is the law of cosines. As for the law of sines, this rule is valid for all triangles regardless of the angles. The law of cosines can be seen below.

For the special case of the right triangle, C = 90º and the third term drops out to give the Pythagorean theorem which does not involve either of the other angles.. The Pythagorean theorem is seen below.

Each triangle has six parts, three sides and three angles. If three of these are known including at least one side, the other three can be calculated using the two laws.

Similar Triangles

Similar triangles are triangles that have the exact same angles as each other but not necessarily the same side lengths. There are certain rules that can be determined about similar triangles. If you superimpose one triangle on the other so that one of the corners and two of the sides match, it can be seen that the third sides of each triangle are parallel to each other. Another rule important later in determining geometric unsharpness in radiology is the ratio of sides. The ratio of the two sides a is the same as the ratio of the two sides b and is also the same as the two sides c. This is illustrated below.

pre-calculus graph

Graph of a rational function can be discontinuities because it has polynomial in the denominator. Is possible value x divide by zero?

Example, there is a function f, it is f of x equals x plus two divided by x minus one. If we insert x equals one, we get the value of f of one equals one plus two divided by one minus one equals three divided by zero. We know that three divided by zero is bad idea and the graph f of x equals x plus two divided by x minus one will break in function graph. If we insert x equals zero, we will get the function f of zero equals zero plus two divided by zero minus one equals negative two.

We will draw graph f of x equals x plus two divided by x minus one in the x y-coordinate plane. First we draw x y-coordinate plane, then we draw the graph f of x equals x plus two divided by x minus one as we know that this graph intersects axis y at point (0, -2), then we can draw the graph which through point (0, -2) and approach axis x in axis positive y. So we can say that graph f of x equals x plus two divided by x minus one is break. On the contrary, if there is line x equals one, so the graph approaches this line and axis positive x is discontinue.

Rational function does not always work this way!

Not all rational functions will give zero in denominator. Take the graph f of x equals one over x square plus one, it’s denominator never zero because of plus one and of course this graph is not break in function graph.

Do not forget!

Rational function denominator can be zero if the polynomial have smooth and unbroken curve and for rational function x equals zero in denominator, because that is impossible situation. It is impossible because there is not value for the function, so make break in function graph.

The break is showed up by two ways!

First, it can break in graph function because of the missing point in the graph. For example the graph f of x equals x square minus x minus six over x minus three has missing point at line x equals three. It can be happened because if we insert x equals three to the graph, so we get f of three equals three square minus three minus three over three minus three equals zero over zero. It can break because the value zero over zero is not possible, not feasible and not allowed. Typical f of three equals three squares minus three minus three over three minus three equals zero over zero named missing point syndrome. If the result is zero over zero, to solve this problem we can make the factor top and button to be simplify. For example, y equals x square minus x minus six over x minus three. We get the top factor are x minus three and x plus two, the button factor is x minus three. So we can write y equals (x minus three) times (x plus two) over x minus three, and we can divide numerator and nominator with x minus three so we get y equals x plus two. Now, it is not problem if we insert x equals three to the new function y equals x plus two.

mathematics

Mathematics is one of the most useful and fascinating divisions of human knowledge. It helps us in many important areas of study, and has the power to solve some of the deepest puzzles man must face.

Mathematics includes many different subjects. So the terms “mathematics” is usually hard to define. But here is a definition that fits most of the mathematics we learn in school or college. Mathematics is the study of quantities and relations through the use of numbers and symbol. Arithmetic, for example, deals with the quantities expressed by numbers. Algebra uses quantities and relations expressed by symbols. Geometry involves quantities associated with figures in space. Trigonometry is concerned with the measurement of angles and with the relationships of angles. Analytic geometry applies algebra to geometric studies. And calculus works with pairs of associated quantities and the way quantity changes in relation to the other.

We use mathematics in our daily life, even in such simple ways as telling time from a clock or counting the change returned by the grocer. A customer in a store uses mathematics whenever he buys something. In science, most scientists depend on mathematics exact descriptions and formulas of observation in experiments. Many scientific problems have become so complicated that only’ highly trained mathematicians working with giant electronic computers can supply the answers.

Mathematics has great importance in engineering projects. For example, the design of super highway requires extensive use of mathematics. The construction of giant dam would be impossible without first filling reams of paper with mathematics formulas and calculations. In business, all transactions that involve buying arid selling call for mathematics. Many companies employ accountants to keep their records and statisticians to analyze large group of figures.

funny math

Trick 1: Number below 10
Step1: Think of a number below 10.
Step2: Double the number you have thought.
Step3: Add 6 with the getting result.
Step4: Half the answer, that is divide it by 2.
Step5: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought.

Answer: 3

Trick 2: Any Number
Step1: Think of any number.
Step2: Subtract the number you have thought with 1.
Step3: Multiply the result with 3.
Step4: Add 12 with the result.
Step5: Divide the answer by 3.
Step6: Add 5 with the answer.
Step7: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought.

Answer: 8

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Trick 3: Any Number
Step1: Think of any number.
Step2: Multiply the number you have thought with 3.
Step3: Add 45 with the result.
Step4: Double the result.
Step5: Divide the answer by 6.
Step6: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought.

Answer: 15

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Trick 4: Same 3 Digit Number
Step1: Think of any 3 digit number, but each of the digits must be the same as. Ex: 333, 666.
Step2: Add up the digits.
Step3: Divide the 3 digit number with the digits added up.

Answer: 37

13. The figure above shows the graph of y=g(x). h(x)=g(2x)+2

h(1)?

Solution

1. We have h(x)=g(2x)+2 …i and h(1)=1….ii

We can substitute 1 to first equation, and we know it is h(1)

For x=1 we have h(1)=g(2)+2…iii

2. We have y=g(x)…iv

The meaning of g(2) is when x=2, then y=1…v

3. From fifth and third equation we have h(1)=1+2

Then h(1) =3

In the mathematic we can write

h(x)=g(2x)+2

h(1)=g(2)+2

h(1)=1+2

h(1) =3

13. Let the function f be defined by f(x)=x+1, if 2f(p)=20. what is the value of f(3p)?

Solution

1. From 2f(p)=20 we can look for p.

If 2f(p)=20 and the equation is over 2 we have f(p)=10

Then we substitute p to f(x). and we have f(p)=p+1

f(p) is 10 and f(p) is p+1

Then p+1=10

And we have p is 10-1

And p is 9

2. If p=9, then f(3p) is 3p+1=(3*9)+1

f(3p) is 27+1

And we have f(3p) is 28

17. in the xy- coordinate plane, the graph of x=y^2 – 4. intersects line l at (0,p) and (5,t). what is the greates possible value of the slope of l?

Basic Trigonometry

Trigonometry ( from trigon and metron )

Trigonometry is really study of rectangle and the relationship between the side and the angle of rectangle.

Sin θ = ?

Cos θ = ?

Tan θ = ?

To solve them use trig. The trig is remember "SOH", "CAH", "TOA"

SOH = Sine is opposite over hypotenuse.

CAH = Cosine is adjacent over hypotenuse.

TOA = Tangent is opposite over adjacent.

Sin θ = opp/hyp

= 4/5

Cos θ = adj/hyp

= 3/5

Tan θ = opp/adj

= 4/3

If the angle is on x, tan x = 3/4.