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An asymptote is basically a line in which a graph approaches, but it never intersects the line.
For example: In the following given graph of P = $\frac{1}{Q}$, (here the x axis is denoted by ‘P’ and y axis is denoted by Q) the line approaches the p-axis (Q = 0), but the line never touches it. The line will not actually reach Q = 0, but will always get closer and closer.
It means the line Q = 0 is along to the horizontal asymptote. We will see in the example that the value Q = 1 / P is a fraction. If we find the infinity on the x-axis then the top of the fraction remains 1, but the bottom fraction gets bigger and bigger. As a result we can say that the entire fraction actually gets smaller, although it will not be zero. The function will be 1/2, then 1/3, then 1/10, even 1/10000, but never quite 0.
An asymptote is a straight line at a finite distance from the origin to which distance a tangent to a curve tends as the distance from the origin of the point of contact tends to infinity.

Horizontal Asymptote 

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A horizontal asymptote is a y-value which a function approaches. The asymptote is parallel to x axis, so is of the form y = a, for all real values of a. Thus if f(x) = y is a function, then y = a, becomes undefined for the function, then the line y = a, is called the asymptote for the function y = f(x).
To find the horizontal asymptote the given function should be in a standard form. Horizontal asymptotes take place as the graph of the function extends forever to the left or to the right.
By this, we are looking for very large positive or negative values of x. If the numerator is less than the denominator then the x axis is the horizontal asymptote.

Vertical Asymptote 

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Vertical asymptotes are vertical lines passing through the zeroes of the denominator of a rational function. Vertical asymptotes are where the denominator of a fraction becomes 0 and the value of f(x) becomes undefined.
Set the denominator to 0 and solve. First, we find the domain of the function and we will find out the points which are not included in the domain. Then we draw vertical lines through those points which are known as vertical asymptotes.

If there is no denominator, then there will be no vertical asymptotes.
Example: Find vertical asymptotes of f(x) = $\frac{x - 9}{x^{2} + 16}$

Solution : For the given equation set the denominator to zero.

x$^{2}$ + 16 = 0
x$^{2}$ = - 16
As there are no real solutions there will be no vertical asymptotes.

Oblique Asymptotes

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If the degree is greater in the numerator than in the denominator then we will have an oblique asymptote. Since the numerator being stronger it is ought to pull the graph away from the x axis. In oblique asymptotes the linear asymptote are not parallel to the coordinates x and y. Oblique asymptote is also called slant asymptote.
For a polynomial
f(x) = $\frac{ax^{n} + ......}{bx^{m} + .....}$
Numerator is a nth degree polynomial and the denominator is a mth degree polynomial.
So when n > m there will be a slant diagonal or oblique asymptote. We use long division to divide the denominator into the numerator.
Linear asymptote is not parallel to either x axis or y axis.
When the degree of numerator is one more than the denominator we use
R(x) = $\frac{P(x)}{Q(x)}$ = f(x) + $\frac{r(x)}{Q(x)}$
$\frac{r(x)}{Q(x)}$ $\rightarrow$ 0 as x $\rightarrow$ $\infty$ or as x $\rightarrow$ -$\infty$
An oblique Asymptote is a line of the form y = mx + b where m $\neq$ 0.

To find Oblique Asymptote of a rational function,
Step 1: Divide the numerator by the denominator using long division or synthetic division.

Step 2: The result of division is a non-fractional part and a fractional part. The non-fractional part is the Oblique Asymptote. The fractional part approaches zero as x tends to negative infinity or positive infinity.

Examples of Asymptotes

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Example 1: Find the oblique asymptote of
R(x) = $\frac{x^{2} + 5x -15}{x +2}$

Solution: Divide the numerator by denominator by using long division so that we get

x + 2 )x2 + 5x - 15(x + 3
x2 + 2x
3x - 15
3x + 6
- 9

The result of R(x) = $\frac{x^{2} + 5x - 15}{x + 2}$ = (x + 3) + $\frac{-9}{x + 2}$
In this case oblique asymptote is y = x + 3

Example 2: Find vertical and horizontal asymptotes for y = $\frac{x^{2} + 9x + 2}{3x^{2} - 12}$
To find vertical asymptote we set the denominator equal to zero and solve.
3x$^{2}$ - 12 = 0
3x$^{2}$ = 12
x = $\pm$ 2
Horizontal asymptote
The highest degree is 2 by dividing the leading terms we get
y = $\frac{x^{2}}{3x^{2}}$

y = $\frac{1}{3}$