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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