Question 310822
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A rational number is a number that can be expressed as the ratio of two integers.  An irrational number cannot be so expressed.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{1278}{34567788}] is a rational number


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 4] is a rational number


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sqrt{3}] is an irrational number.  It is also an algebraic number (it is the root of a non-constant polynomial equation with rational coefficients).


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \pi] is a transcendental irrational number (it is irrational AND it is NOT the root of any non-constant polynomial equation with rational coefficients)


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{32727272727}{100000000000}]



*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ \frac{2x\ +\ 1}{(x\ +\ 2)^3}] is a rational function.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ g(x)\ =\ \alpha_0x^n\ +\ \alpha_1x^{n-1}\ +\ \alpha_3x^{n-2}\ +\ \cdots\ +\ \alpha_{n-2}x^2\ +\ \alpha_{n-1}x\ +\ \alpha_n] is an *[tex \Large n\text{th}] degree polynomial equation if *[tex \Large \alpha_0\ \neq\ 0] 


A hole is a hole in a graph. That is, a discontinuity that can be "repaired" by filling in a single point. In other words, a removable discontinuity is a point at which a graph is not connected but can be made connected by filling in a single point.


Formally, a removable discontinuity is one at which the limit of the function exists but does not equal the value of the function at that point; this may be because the function does not exist at that point.


Essential Discontinuity

Any discontinuity that is not removable. That is, a place where a graph is not connected and cannot be made connected simply by filling in a single point. Step discontinuities and vertical asymptotes are two types of essential discontinuities.

Formally, an essential discontinuity is a discontinuity at which the limit of the function does not exist.



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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