Question

If the graph of a rational function $R$has the vertical asymptote $x=4$, the factor $x-4$must be present in the denominator of $R$. Explain why.

Short answer

The factor $\left(x-4\right)$must be in the denominator of the$R$. This is because, If we will put the value of$x=4$in that function, then$R\to \infty$that's why$\left(x-4\right)$must be present in the denominator being its vertical asymptote.

Step-by-step solution

Step 1 Definition for the asymptote:

An asymptote is a straight line that continually approaches a given curve but does not meet it at any finite distance.

They are of three types. They are,

1. Vertical asymptote.
2. Horizontal asymptote.
3. Oblique asymptote.

Step 2 we know that for the vertical asymptote limy→∞fx.

So, if the function is of the form$\frac{p}{q}$, then $q$has to be zero for $\underset{y\to \infty }{\lim }f\left(x\right)$so as in question, it is given that $x=4$is a vertical asymptote.

This shows that $\left(x-4\right)$must be in the denominator of the function $R$because if we will put the value of $x=4$in that function then $y\to \infty$, that's why $\left(x-4\right)$must be present in the denominator for $x=4$being its vertical asymptote.