Find the Asymptotes f(x)=2 log base 3 of x+1-3

Find where the expression ${\displaystyle 2{\log }_3(x+1)-3}$ is undefined.

Since ${\displaystyle 2{\log }_3(x+1)-3}$${\displaystyle \to }$${\displaystyle \infty }$ as ${\displaystyle x}$${\displaystyle \to }$${\displaystyle -1}$ from the left and ${\displaystyle 2{\log }_3(x+1)-3}$${\displaystyle \to }$${\displaystyle -\infty }$ as ${\displaystyle x}$${\displaystyle \to }$${\displaystyle -1}$ from the right, then ${\displaystyle x=-1}$ is a vertical asymptote.

Ignoring the logarithm, consider the rational function ${\displaystyle R\left(x\right)=\frac{a{x}^n}{b{x}^m}}$ where ${\displaystyle n}$ is the degree of the numerator and ${\displaystyle m}$ is the degree of the denominator.

1. If ${\displaystyle n<m}$, then the x-axis, ${\displaystyle y=0}$, is the horizontal asymptote.

2. If ${\displaystyle n=m}$, then the horizontal asymptote is the line ${\displaystyle y=\frac{a}{b}}$.

3. If ${\displaystyle n>m}$, then there is no horizontal asymptote (there is an oblique asymptote).

There are no horizontal asymptotes because ${\displaystyle Q\left(x\right)}$ is ${\displaystyle 1}$.

No Horizontal Asymptotes

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

This is the set of all asymptotes.

Vertical Asymptotes: ${\displaystyle x=-1}$

No Horizontal Asymptotes