Vertical asymptote concept used to solve this question. First, I explain the concept, and then, you use it to solve the question.
Using this, we get that the correct option is:
There is a vertical asymptote at x = –5, because limit of g(x) as x approaches -5 minus = negative infinite and limit of g(x) as x approaches -5 plus, it is positive inifnite.
Vertical asymptote:
A vertical asymptote is a value of x for which the function is not defined, that is, it is a point which is outside the domain of a function;
In a graphic, these vertical asymptotes are given by dashed vertical lines.
An example is a value of x for which the denominator of the function is 0, and the function approaches infinite for these values of x.
The fraction is:
[tex]g(x) = -\frac{7(x-5)^2(x+6)}{(x-5)(x+5)}[/tex]
Simplifying:
The term (x-5) is present both at the numerator and at the denominator, and thus, the function can be simplified as:
[tex]g(x) = -\frac{7(x-5)(x+6)}{x+5}[/tex]
Vertical asymptote:
Point in which the denominator is 0, so:
[tex]x + 5 = 0[/tex]
[tex]x = -5[/tex]
Now, we take a look at the graphic of the simplified function, given at the end of this answer.
You can see that as x approaches -5 to the left, that is -5 minus, the function goes to minus infinite, and as x approaches -5 to the right, that is, -5 plus the function goes to plus infinite, and thus, the correct answer is:
There is a vertical asymptote at x = –5, because limit of g(x) as x approaches -5 minus = negative infinite and limit of g(x) as x approaches -5 plus, it is positive inifnite.
For another example of vertical asymptotes, you can check https://brainly.com/question/24278113
