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Horizontal Asymptote


Asymptote of a curve can be defined as a line in such a way that the distance between the curve and the line reaches to value zero, as they tends to infinity. In coordinate Geometry, there are two type of asymptotes.

Horizontal Asymptote: Horizontal Asymptotes are the horizontal lines in which graph of function tend to $a \rightarrow \pm \infty$.

The horizontal line is given by a = c. The given equation is a horizontal asymptote of a function a = f (t).

If it satisfies the given equation, then

$\lim_{t \to -\infty} f (t) = c$

Or, we can write it as:

$\lim_{t \to \infty} f (t) = c$

Now, we will see how to solve horizontal asymptotes. Take an example and see how to solve horizontal asymptote.

Let f (t) = $\frac{(2t - 1)(t + 3)}{t(t - 2)}$

As we know that the given function is in factor form, first we will convert the given function in to standard form. To find standard form, we have to multiply the given values. So, the standard form of the equation is as follows:

f(t) = $\frac{2t^2 + 5t - 3}{t^2 - 2t}$

From the given equation, we neglect every value except the biggest exponents of ‘t’ which is present in the numerator and the denominator.

So, we can write it as follows:

f (t) = $\frac{2t^2}{t^2}$

On solving, we get 2.

So, the horizontal asymptote for the horizontal line is y = 2.

Vertical Asymptote: The equation of the vertical line is given by x = t

Given equation is a vertical asymptote of a graph which has a function y = f (t). This function is applicable, when one of the given condition is true.

The two conditions are shown below:

$\lim_{t \to a-} f (t) = \pm \infty$
$\lim_{t \to a+} f (t) = \pm \infty$

This is a brief introduction of horizontal asymptote and vertical asymptote.

Horizontal Asymptote Rules

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In coordinate Geometry, an asymptote of a curve is generally a straight line, that approaches the curve so close, that it tends to meet the curve at infinite Position. But, it never intersects the line.

There are two types of asymptotes as follows:
  1. Vertical asymptotes
  2. Horizontal asymptotes

If the graph of a function approaches as $x \to \pm \infty$ for horizontal lines, then it is horizontal asymptotes.

The horizontal line is given by:

u = d. Given equation is a Horizontal Asymptote of a function u = f (q).

If it satisfies the given equation,

$\lim_{q \to -\infty} f (q) = d$

We can write it as:

$\lim_{q \to +\infty} f (q) = d$

Let us discuss the Horizontal Asymptote rules.

Horizontal Asymptote Rules:

Rule 1:
If the degree of the numerator is larger than the degree of the denominator, then there will be no horizontal asymptotes. Even if it is larger by exactly 1, there can be a slant asymptote.

Rule no.2: If degree of the numerator and degree of the denominator are equal, then the horizontal asymptote will be fraction of leading coefficients.

Rule no. 3: If degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote must be a line y = 0.