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Last modified on August 28th, 2023

An asymptote is a straight line or a curve that approaches a given curve as it heads toward infinity but never meets the curve. Such a pair of curves is called an asymptotic curve.

Asymptotes characterize the graphs of rational functions ${f\left( x\right) =\dfrac{P\left( x\right) }{Q\left( x\right) }}$ , here p(x) and q(x) are polynomial functions.

Mathematically, an asymptote of the curve y = f(x) or in form f(x, y) is a straight line such that the distance between the curve and the straight line tends to zero as both approach infinity.

A typical example of asymptotes is vertical and horizontal lines given by x = 0 and y = 0, respectively, relative to the graph of the real-valued function ${f\left( x\right) =\dfrac{1}{x}}$ in the first quadrant. If we notice, ${\lim_{x\rightarrow 0}\dfrac{1}{x}=\alpha}$and ${\lim _{x\rightarrow \alpha }\dfrac{1}{x}=0}$

Since, ${\dfrac{1}{0}}$ is undefined and there exists no real number x such that ${\dfrac{1}{x}=0}$, the straight lines x = 0, y = 0 never intersects the curve and thus ${f\left( x\right) =\dfrac{1}{x}}$ represents an asymptote as shown below.

The asymptote can approach from any side, may cross over, or even move away and back again. But the curve never overtakes the asymptote. Based on how it approaches the curve, there are 3 types of asymptotes:

The asymptote is a horizontal asymptote when **x** tends to infinity or â€“infinity, and the curve approaches some constant value **b**.

The asymptote is a vertical asymptote when **x** approaches some constant value **c** from left to right, and the curve tends to infinity or -infinity.

The asymptote is an oblique or slant asymptote when x moves towards infinity or â€“infinity and the curve moves towards a line y = mx + b. Here, m is not zero as in horizontal asymptote.

Asymptotes can also approach infinity and can be negative.

Since an asymptote is a horizontal, vertical, or slanting line, its equation is x = a, y = a, or y = ax + b. We can find the different types of asymptotes of a function y = f(x).

The horizontal asymptote, for the graph function y=f(x), where the equation of the straight line is y = b, which is the asymptote of a function${x\rightarrow +\alpha }$, if the given limit is finite:

${\lim_{x\rightarrow +\alpha }f\left( x\right) =b}$

The above limit is the same for ${x\rightarrow-\alpha }$

The vertical asymptote has a straight line equation x = a for the function y = f(x) if it satisfies either of the two conditions:

${\lim_{x\rightarrow a+0}f\left( x\right) =\pm \alpha}$

or

${\lim_{x\rightarrow a-0}f\left( x\right) =\pm \alpha}$

Else, at least one of the one-sided limits at point x=a must equal infinity.

The oblique asymptote is any line of the form y = mx + b for some real numbers m â‰ 0 and b such that a curve given by the function y = f(x) approaches but never intersects the straight line, for the limit ${x\rightarrow +\alpha }$, only if the given limits are finite:

${\lim_{x\rightarrow +\alpha }\dfrac{f\left( x\right) }{x}=k}$

${\lim_{x\rightarrow \alpha }\left[ f\left( x\right) -kx\right] =b}$

**Find all the asymptotes of the function ${f\left( x\right) =\dfrac{x^{2}-x+4}{2x+2}}$.**

Solution:

For ${f\left( -1\right) =\dfrac{\left( -1\right) ^{2}-\left( -1\right) +4}{2\left( -1\right) +2}=\dfrac{6}{0}}$ and thus f is undefined.

Again, ${\lim_{x\rightarrow -1}+g\left( x\right) =\alpha}$ and ${\lim _{x\rightarrow -1}-g\left( x\right) =-\alpha}$

Thus, f(x) has a vertical asymptote of x = -1. Also, since â€˜fâ€™ is a rational function and the degree of the numerator is 1 more than the degree of the denominator, the graph of â€˜fâ€™ will give an oblique asymptote.

We can use the polynomial long division method to find the oblique asymptote.

${f\left( x\right) =\dfrac{x^{2}-x+4}{2x+2}}$

= ${\dfrac{x}{2}-\dfrac{3}{x+1}-1}$

Thus, f(x) represents an oblique asymptote ${y=\dfrac{x}{2}-1}$ as ${\lim _{x\rightarrow \alpha }\dfrac{3}{x+1}=0}$ and ${\lim _{x\rightarrow -\alpha }\dfrac{3}{x+1}=0}$

**Calculate the horizontal asymptote of the function ${f\left( x\right) =\dfrac{4x^{3}+3x}{5-3x^{3}}}$.**

Solution:

Here, the highest power in both numerator and denominator is the same, that is, 3

So, the graph of this function will have an asymptote which is the value of the division of the coefficients of the terms with the highest powers. Those coefficients are 4 and âˆ’3.

Thus, the horizontal asymptote is ${b\left( x\right) =-\dfrac{4}{3}}$

**Graph the vertical and horizontal asymptotes of the rational function ${f\left( x\right) =\dfrac{3}{-x+4}}$**

Solution:

Given, ${f\left( x\right) =\dfrac{3}{-x+4}}$

To find vertical asymptotes, we need to make the denominator zero and then solve for x

Here, when x = 4 the denominator = 0 so the vertical asymptote is x = 4

To find the horizontal asymptote, we find the highest power (degree) of the numerator and denominator of the function f(x)

Here, the degree of the numerator is 3, and the degree of the denominator is 4

As the degree of the numerator < the degree of the denominator, the horizontal asymptote is y = 0

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Last modified on August 28th, 2023