Q.33 - 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.

$f\left(x\right)={\tan }^{-1}\left(3x\right)+1$

Step-by-Step Solution

Answer

The function has a real root at $-\frac{\tan \left(1\right)}{3}$

The function has no discontinuity.

The horizontal asymptotes of the function are $y=-\frac{\mathrm{\pi }}{2}+1,y=\frac{\mathrm{\pi }}{2}+1$.

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.

$f\left(x\right)={\tan }^{-1}\left(3x\right)+1$

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:

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

Hence the root of the function is $-\frac{\tan \left(1\right)}{3}$

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 inverse tangent function is a continuous function and has no discontinuities.

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 $x\to -\infty$,

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

Hence one horizontal asymptote is $y=-\frac{\mathrm{\pi }}{2}+1$

When $x\to \infty$,

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

Hence another horizontal asymptote is $y=-\frac{\mathrm{\pi }}{2}+1$

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 a real root at $-\frac{\tan \left(1\right)}{3}$.

The function has no discontinuity.

The horizontal asymptotes of the function are $y=-\frac{\mathrm{\pi }}{2}+1,y=\frac{\mathrm{\pi }}{2}+1$.

The function has no vertical asymptote.