Answer: The Horizontal asymptote of f(x) and g(x) is y=0. The third option is correct.
Explanation:
it is given that the using the rules of transformations to compare the graphs of the functions,
![f(x)=\frac{1}{x}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B1%7D%7Bx%7D)
![g(x)=\frac{5}{x-1}](https://tex.z-dn.net/?f=g%28x%29%3D%5Cfrac%7B5%7D%7Bx-1%7D)
The graph of f(x) vertically stretch by factor 5 and shifts 1 unit right to transform g(x).
To find vertical asymptotes equate denominator equal to 0.
![x=0](https://tex.z-dn.net/?f=x%3D0)
![x=1](https://tex.z-dn.net/?f=x%3D1)
Therefore the function f(x) has vertical asymptote x=0, and g(x) has vertical asymptote x=1. So their is not common vertical asymptote.
To find horizontal asymptotes put
.
![f(x)\rightarrow 0\text{ as }x\rightarrow \infty](https://tex.z-dn.net/?f=f%28x%29%5Crightarrow%200%5Ctext%7B%20as%20%7Dx%5Crightarrow%20%5Cinfty)
![g(x)\rightarrow 0\text{ as }x\rightarrow \infty](https://tex.z-dn.net/?f=g%28x%29%5Crightarrow%200%5Ctext%7B%20as%20%7Dx%5Crightarrow%20%5Cinfty)
Therefore, the function have common horizontal asymptote, i.e., y=0.