Answer:
[tex]\mathrm{Vertical}:\:x=0,\:\mathrm{Horizontal}:\:y=-1[/tex]
The graph of the function is also attached.
Step-by-step explanation:
Considering the expression
[tex]\:f\left(x\right)=\left(\frac{5}{x}+7\right)-8\:\:[/tex]
[tex]\mathrm{Simplify}\:\frac{5}{x}+7-8:\quad \frac{5}{x}-1[/tex]
[tex]\mathrm{Vertical\:asymptotes\:of\:}\frac{5}{x}-1:[/tex]
Go over every undefined point x = a and check if at least one of the following statement is true.
[tex]\lim _{x\to a^-}f\left(x\right)=\pm \infty[/tex]
[tex]\lim _{x\to a^+}f\left(x\right)=\pm \infty[/tex]
[tex]\mathrm{Take\:the\:denominator\left(s\right)\:of\:}\frac{5}{x}-1\mathrm{\:and\:compare\:to\:zero}[/tex]
[tex]x=0[/tex]
The following points are undefined
[tex]x=0[/tex]
[tex]\mathrm{The\:vertical\:asymptotes\:are:}[/tex]
[tex]x=0[/tex]
[tex]\mathrm{Horizontal\:Asymptotes\:of\:}\frac{5}{x}-1:[/tex]
[tex]\mathrm{Check\:if\:at\:}x\to \pm \infty \mathrm{\:the\:function\:}y=\frac{5}{x}-1\mathrm{\:behaves\:as\:a\:line,\:}y=mx+b[/tex]
[tex]\mathrm{Find\:an\:asymptote\:for\:}x\to -\infty \::[/tex]
[tex]\lim _{x\to -\infty \:}\frac{f\left(x\right)}{x}=\lim _{x\to -\infty \:}\frac{\frac{5}{x}-1}{x}=0[/tex]
[tex]\lim _{x\to -\infty \:}f\left(x\right)-mx=\lim _{x\to -\infty \:}\frac{5}{x}-1-0x=-1[/tex]
[tex]y=-1[/tex]
[tex]\mathrm{Find\:an\:asymptote\:for\:}x\to \infty:[/tex]
[tex]\lim _{x\to \infty \:}\frac{f\left(x\right)}{x}=\lim _{x\to \infty \:}\frac{\frac{5}{x}-1}{x}=0[/tex]
[tex]\lim _{x\to \infty \:}f\left(x\right)-mx=\lim _{x\to \infty \:}\frac{5}{x}-1-0x=-1[/tex]
[tex]y=-1[/tex]
[tex]\mathrm{The\:horizontal\:asymptote\:is:}[/tex]
[tex]y=-1[/tex]
Therefore,
[tex]\mathrm{Vertical}:\:x=0,\:\mathrm{Horizontal}:\:y=-1[/tex]
The graph of the function is also attached.
