Chapter 6.5, Problem 2AYU

True or False If $x=3$ is a vertical asymptote of a rational function $R$, then $\underset{x\to 3}{\lim }\left|R\left(x\right)\right|=\infty$. (pp. 224-227)

To determine

To find: $\underset{x\to 3}{\lim }\left|R\left(x\right)\right|\text{ = }\infty$

Answer to Problem 2AYU

Solution:

$\underset{x\to 3}{\lim }\left|R\left(x\right)\right|\text{ = }\infty$, is not true.

Explanation of Solution

Given:

If $x=3$ is a vertical asymptote of a rational function $R$.

Calculation:

It is given that $x=3$ is a vertical asymptote of a rational function $R$ it means that function is not defined at $x=3$ or its denominator term becomes zero on plug in $x=3$ in it, taking the form $\frac{1}{0}$ but it may be possible that while solving $R\left(x\right)$, denominator term got cancelled with some factor of numerator term, that would given a real value of $R\left(x\right)$.

For example :

$\underset{x\to 3}{\lim }\left|R\left(x\right)\right|=\frac{x^2-9}{x-3}$

$=\frac{\left(x-3\right)\left(x+3\right)}{x-3}$

$=x+3$

$=6$

Here it is clear that the answer is being 6.

So given equation $\underset{x\to 3}{\lim }\left|R\left(x\right)\right|\text{ = }\infty$, is not true or its false.