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Fast Formula for Limits

One of the math topics that are always there in the entrance exam State Universities in Indonesia (PTN) since Skalu until SNMPTN is about the limit.
For that purpose I wrote a formula which could hopefully be useful to work on the problems about the limits more quickly.

No	Fast formula
1	Limit f(x) for $x\rightarrow0$ $\lim_{x\to 0}\frac{dx}{\sqrt{ax+b}-\sqrt{cx+b}}=\frac{2\cdot d\cdot \sqrt{b}}{a-c}$ $\lim_{x\to 0}\frac{\sqrt{ax+b}-\sqrt{cx+b}}{dx}=\frac{a-c}{2\cdot d\cdot \sqrt{b}}$
2	Limit F(x) for $x\rightarrow \infty$ , fractional form $p>r\Rightarrow \lim_{x\to \infty }\frac{ax^{p}+bx^{q}}{cx^{r}+dx^{s}}=\infty$ $p=r\Rightarrow \lim_{x\to \infty }\frac{ax^{p}+bx^{q}}{cx^{r}+dx^{s}}=\frac{a}{c}$ $p<r\Rightarrow \lim_{x\to \infty }\frac{ax^{p}+bx^{q}}{cx^{r}+dx^{s}}=0$
3	Limit F(x) for $x\rightarrow \infty$ , quadratic form $a>p\Rightarrow \lim_{x\rightarrow \infty }\sqrt{ax^{2}+bx+c}-\sqrt{px^{2}+qx+r}=+\infty$ $a=p\Rightarrow \lim_{x\rightarrow \infty }\sqrt{ax^{2}+bx+c}-\sqrt{px^{2}+qx+r}=\frac{b-r }{2\sqrt{a}}$ $a<p\Rightarrow \lim_{x\rightarrow \infty }\sqrt{ax^{2}+bx+c}-\sqrt{px^{2}+qx+r}=-\infty$
4	Trigonometric Limits Result limit = the product of the numerator coefficients x divided by the product of the coefficient of x the denominator $\lim_{x\rightarrow 0 }\frac{\sin ax\cdot \tan bx}{px\cdot \sin cx}=\frac{ab}{pc}$
5	Limits of trigonometric form (1 - cos ax) $\lim_{x\rightarrow 0 }\frac{1-\cos ax}{bx\cdot \tan cx}=\frac{a^{2}}{2bc}$ $\lim_{x\rightarrow 0 }\frac{bx\cdot \tan cx}{1-\cos ax}=\frac{2bc}{a^{2}}$

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Write UM UGM exam was held to select new students qualified candidates with superior academic ability and potential to support the completion of education at University of Gajah Mada (UGM) appropriate time limit has been set. Material selection includes Potential Test, Basic Sciences, Natural Sciences, and Social Sciences. Examination held in Yogyakarta, Semarang, Madiun, Jakarta, Tangerang, Cirebon, Palembang, Pekanbaru, Balikpapan, Lhokseumawe, Bandar Lampung, Batam, Denpasar and Mataram.

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UM-UGM 2010 – Kemampuan IPA kode 452

Fast formula for mathematics

When going to take the exam in State Universities (PTN), you must prepare yourself with the best, to understand the concept of the tested subjects. You also need to have quick ways to solve problems, especially for math problems.
Here are examples of quick formulas to solve math problems that are usually discussed in the Test Guidance institutions.

No	Fast formula
1	Composition functions $f_{o}g\left ( x \right )=h\left ( x \right )\Rightarrow f\left ( x \right )=h\left ( g^{-1}\left ( x \right ) \right )$
2	Inverse function of a linear $f\left ( x \right )=ax+b\Rightarrow f^{-1}\left ( x \right )=\frac{x-b}{a}$
3	Inverse function of fractional $f\left ( x \right )=\frac{ax+b}{cx+d}\Rightarrow f^{-1}\left ( x \right )=\frac{-dx+b}{cx-a}$
4	Inverse function of the square root $f\left ( x \right )=\sqrt[n]{ax+b}\Rightarrow f^{-1}\left ( x \right )=\frac{x^{n}-b}{a}$
5	Inverse function of the exponent $f\left ( x \right )=a^{x}\Rightarrow f^{-1}\left ( x \right )=^{a}\log x$
6	Inverse function of the logarithm $f\left ( x \right )=^{a}\log x\Rightarrow f^{-1}\left ( x \right )=a^{x}$
7	Inverse functions and composition $\left ( f_{o}g \right )^{-1}=\left g^{-1} \right_{o}f^{-1}\left ( x \right )$

STOIKIOMETRI

A. Hukum-hukum dasar Kimia :

Hukum Lavoisier (hukum kekekalan massa) :

$Massa A =\frac{yxArA}{MrA_{y}B_{z}}\times massa A_{y}B_{z}$

$Massa B =\frac{zxArB}{MrA_{y}B_{z}}\times massa A_{y}B_{z}$

$^{o}/_{o}A =\frac{yxArA}{MrA_{x}B_{y}}\times 100^{o}/_{o}$

$^{o}\textrm{/}_{o}B=\frac{zxArB}{MrA_{x}B_{y}}\times 100^{o}\textrm{/}_{o}$

Hukum Dalton (Hukum Perbandingan berganda)
“Jika dua jenis unsur dapat membentuk dua jenis atau lebih persenyawaan, maka perbandingan massa unsur yang terikat dengan massa yang sama, unsur yang satu lagi merupakan bilanga bulat dan sederhana’

Hukum Gay Lussac (Hukum Perbandingan Volume)
“Bila diukur pada suhu dan tekanan yang sama, volume gas yang bereaksi dan gas hasil reaksi berbanding sebagai bilangan bulat dan sederhana”
Untuk reaksi yang melibatkan gas – gas digunakan hukum ini

Hipotesis Avogadro
“Pada suhu dan tekanan yang sama semua gas bervolume sama mengandung jumlah molekul yang sama pula”.

B. Massa Atom Relatif (Ar) Dan Massa Molekul Relatif (Mr)

Massa atom relative (Ar) suatu unsure dan massa molekul relative (Mr) suatu senyawa didasarkan pada massa atom isotop C-12. hal ini didasarkan karena isotop C-12 adalah isotop paling stabil.
Ar dan Mr dirumuskan sebagai berikut :

${\color{blue} ArX=\frac{massa 1 atom X}{\frac{1}{12}\times massa 1 atom C^{12}}}$

${\color{blue} MrXY=\frac{massa 1 molekul XY}{\frac{1}{12}\times massa 1 atom C^{12}}}$

Sejalan dengan perkembangan alat spectrometer massa, Ar suatu unsure ditentukan pada massa atom isotop unsure tersebut dan kelimpahannya di alam, maka Ar suatu unsure dapat ditentukan dengan

${\color{blue} Ar=\frac{m_{1}A_{1}+m_{2}A_{2}+ .... +m_{x}A_{x}}{100}}$

(mohon maaf postingan ini belum selesai, maklum baru belajar)

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