Let g(x) = (3x^2 + 14x - 5)/(x^2 - 25)

a)
\(lim_{x\rightarrow-\infty}\frac{3x^2+14x-5}{x^2-25}\\ =lim_{x\rightarrow-\infty}\frac{x^2(3+\frac{14}{x}-\frac{5}{x^2})}{x^2(1-\frac{25}{x^2})}\\ =lim_{x\rightarrow-\infty}\frac{3+\frac{14}{x}-\frac{5}{x^2}}{1-\frac{25}{x^2}}\\ =lim_{x\rightarrow-\infty}\frac{3+0-0}{1-0}=3\)

b)

horizontal asymptotes happen when the limit as x approaches infinity or negative infinity approaches some number

as found above, a horizontal asymptote is y=3

you can also check the limit as x approaches positive infinity as sometimes there are 2 horizontal asymptotes, but in this case that limit also equals 3, so there's only one horizontal asymptote.

c)

\(\frac{3x^2+14x-5}{x^2-25}=\frac{(x+5)(3x-1)}{(x-5)(x+5)}\\ \)

first step is to factor the numerator and denominator

then you see that x+5 can be canceled out. this means that there's a hole (removeable discontinuity) at x=-5

after canceling out you're left with:
\(\frac{3x-1}{x-5}\)

to find the coordinates of the hole plug in x=-5 into that and you get y=8/5, so the hole is at (-5,8/5)

 

to find vertical asymptotes, use the equation that's left over after the canceling and set the denominator equal to 0

x-5 = 0, x = 5
A vertical asymptote is x=5