How to get asymptotes of Rational Function

Asymptotes

There are 2 kinds of asymptotes, vertical and horizontal ones.

Vertical Asymptote

Mathematically any denominator cannot be zero. If you know the x value which makes the denominator zero, that is an asymptote.

Example 1

$$y=\frac{2x-1}{2x+6}-4$$

The denominator is $$2x+6$$

Set $$2x+6=0$$

Solve for x

$$x=-3$$

This is the vertical asymptote of this function.

Horizontal Asymptote

When $$x→∞$$, the graph would approach to the horizontal asymptote. So we should deal with a limit.

Let’s think of the situation that the value of the x approaches to ∞.

Example 2

$$\lim_{x\to\infty} y=\frac{2x-1}{2x+6}-4$$

But we have to arrange this fraction to deal with ∞

Divide by the x which has the largest exponent in the fraction. In this case x is good.

$$y=\frac{2x-1}{2x+6} \frac{(\frac{1}{x})}{(\frac{1}{x})} -4$$

$$y=\frac{2-\frac{1}{x}}{2+\frac{6}{x}}-4$$

Take the limit

$$\lim_{x\to\infty} y=\frac{2-\frac{1}{x}}{2+\frac{6}{x}}-4$$

Since $$\lim_{x\to ∞} \frac{1}{x}=0$$

The equation is going to be

$$y=\frac{2}{2}-4$$

$$y=-3$$

This calculation means the graph approaches to the line $$y=-3$$, but never touch it.

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