Given the function
[tex]R(x)=\frac{7x+7}{6x+12}[/tex]we have to determine the behavior of the graph on either sides of the vertical asymptotes. if one exists.
Remember that vertical asymptotes occur when in a rational function, that is a function of the form
[tex]R(x)=\frac{f(x)}{g(x)}[/tex]The denominator, g(x), becomes 0.
In our case, the denominator of the rational function is
[tex]g(x)=6x+12[/tex]therefore
[tex]6x+12=0\text{ }\Rightarrow\text{ }x=-2[/tex]we conclude that there is a vertical asymptote at x=-2.
Now we will determine the behavior on both sides of the asymptote
One possible approach to discover this behavior is to pick up close to x=-2 values on both sides and evaluate it to determine the sign. Let us take
[tex]x=-2.01\text{ and }x=-1.99[/tex]Evaluation the first one
[tex]R(-2.01)=\frac{7(-2.01)+7}{6(-2.01)+12}=117.83[/tex]since the result is big and positive we conclude that on the left side of the asymptote the function approach to + infinity.
Now we will evaluate on the other side to determine the sign
[tex]R(-1.99)=\frac{7(-1.99)+7}{6(-1.99)+12}=-115.5[/tex]Therefore, the on the right the function approach to - infinity as the numbers approach to x=-2
Looking at the options we see that the correct option is the option c).
