Q10. - Algebra 1

Question

State whether each sentence is true or false. If false, replace the underlined word, phrase, expression, or number to make a true sentence.

 

The rational function $y=\frac{2}{x-1}+3$ has a horizontal asymptote at $\underset{¯}{y=3}$.

Step-by-Step Solution

Answer

The given statement is True.

Step 1. State the concept of the horizontal asymptote.

For the function $y=f\left(x\right)$ if the value of either $\underset{x\to \infty }{\lim }f\left(x\right)=L$ or $\underset{x\to -\infty }{\lim }f\left(x\right)=L$ and L is a finite number, then $y=L$ is known as horizontal asymptote of the given function.

Step 2. Explanation of the statement.

Now, in the given equation $y=\frac{2}{x-1}+3$ to find the value of horizontal asymptote, take limits as:

$\begin{array}{c}\underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\left(\frac{2}{x-1}+3\right)=\frac{2}{\infty -1}+3=0+3=3\\ \underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\left(\frac{2}{x-1}+3\right)=\frac{2}{-\infty -1}+3=0+3=3\end{array}$

Step 3. Write whether the statement is true or false.

Since $\underset{x\to \infty }{\lim }f\left(x\right)=3$ and $\underset{x\to -\infty }{\lim }f\left(x\right)=3$, so $y=3$ is a horizontal asymptote of the given function. Therefore, the given statement “The rational function $y=\frac{2}{x-1}+3$ has a horizontal asymptote at $\underset{¯}{y=3}$.” is true.