Chapter 7.5, Problem 38PPS

To determine

To calculate the equation of horizontal asymptote for the given function.

Answer to Problem 38PPS

  $y=0$

Explanation of Solution

Given:

Function: $y={\left(\frac{2}{3}\right)}^{x-1}$

Concept used:

To find the equation of horizontal asymptotes:

If the degree of the denominator is bigger than the degree of the numerator, the horizontal asymptote is the x -axis.

If the degree of the numerator is bigger than the denominator, there is no horizontal asymptote.

If the degrees of numerator and the denominator are the same, the horizontal asymptote is the leading coefficient of the numerator divided by the leading coefficient of the denominator.

Also, a function $f(x)$ has a horizontal asymptote at $y=k$ if either of the two limit statements is true:

  $\underset{x\to \infty }{\lim }f(x)=k$ and $\underset{x\to -\infty }{\lim }f(x)=k$

Calculation:

The given function is $y={\left(\frac{2}{3}\right)}^{x-1}$

Let $f(x)=y$

Substitute $f(x)$ in the above formula,

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

So, the equation of the horizontal asymptote is $y=0.$

Conclusion:

Therefore, the equation of the horizontal asymptote is $y=0.$