Q.30 - Calculus

Question

Find the roots, discontinuities, and horizontal and vertical asymptotes of the functions in Exercises 23–34. Support your answers by explicitly computing any relevant limits.

$f\left(x\right)=\frac{1}{2+3^x}$

Step-by-Step Solution

Answer

The function has no real roots.

The function has no discontinuity.

The horizontal asymptotes of the function are $y=0,y=\frac{1}{2}$.

The function has no vertical asymptote.

Step 1. Given Information

Given to determine the roots, discontinuities, and horizontal and vertical asymptotes of the functions below.

$f\left(x\right)=\frac{1}{2+3^x}$

Step 2. Roots of a function

The roots of a function are the values of x where the simplified function value is 0.

Simplifying the function and finding the root:

$$\begin{gathered} f\left(x\right)=0 \\ \frac{1}{2+3^x}=0 \end{gathered}$$

Since the denominator is always positive, the expression can never be 0. So, there are no real roots.

Step 3. Discontinuities of a function

A function has a discontinuity at a point $x=a$ where $f\left(a\right)\ne \underset{x\to a}{\lim }f\left(x\right)$.

The function is undefined when the denominator is 0.

Here,

$$\begin{gathered} 2+3^x=0 \\ {3}^x=-2 \end{gathered}$$

Since $3^x$ cannot be negative, the function has no discontinuity.

Step 4. Horizontal asymptote of a function

Horizontal asymptote of a function can be determined by the tangents at the infinity i.e. when x is positive or negative infinity.

When $x\to -\infty$,

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

Hence one horizontal asymptote is $y=\frac{1}{2}$.

When $x\to \infty$,

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

Hence another horizontal asymptote is $y=0$

Step 5. Vertical asymptote of a function

The function has no discontinuity (undefined points). Hence there are no vertical asymptotes.

Step 6. Conclusion

The function has no real roots.

The function has no discontinuity.

The horizontal asymptotes of the function are $y=0,y=\frac{1}{2}$.

The function has no vertical asymptote.