Question

Find the intercepts and asymptotes, use limits to describe the behavior at the vertical asymptotes. (37 and 39)

Answer 1

39)
Its always best to factorise where you can, in most questions it tends to be a necessary step and even if its not, it usually makes things easier;
In this case, you can factorise the denominator:
$h(x) = \frac{x \ - \ 1}{ x^{2} \ - \ x \ - \ 12} \\\\ = \frac{x \ - \ 1}{(x \ - \ 4)(x \ + \ 3)}$

To get y-intercept/s, set x = 0 and solve:
$h(0) = \frac{(0) \ - \ 1}{((0) \ - \ 4)((0) \ + \ 3)} \\\\ = \frac{-1}{-12} \\\\ = \frac{1}{12}$
y-intercept: (0, 1/12)

To get x-intercept/s, set y = 0 and solve:
$\frac{x \ - \ 1}{(x \ - \ 4)(x \ + \ 3)} = 0 \\\\ x \ - \ 1 = 0 \\\\ x = 1$
x-intercept: (1, 0)

To find the asymptotes, it is necessary for us to have factorised the denominator so its a good thing that I did so at the beginning;

To get the vertical asymptote/s, set the denominator equal to 0, then just rearrange to get x:
$(x \ - \ 4)(x \ + \ 3) = 0 \\ x \ - \ 4 = 0 \\ x = 4 \\ and \\ x \ + \ 3 = 0 \\ x = -3$
vertical asymptotes: x = 4  and  x = -3

There are no horizontal asymptotes.