I am assuming this in not in a Calculus class where you have discussed derivatives. If you are familiar with derivatives, let me know and I will offer a solution using derivatives.
For a rational function a vertical asymptote occurs when a value of x makes the denominator 0 but not the numerator. Let's find that value of x.
x + 4 = 0 => x = -4
Let's see if x = -4 causes the numerator to be zero.
3*(-4) - 1 = -13 ≠ 0
So x = -4 is a vertical asymptote.
To find the horizontal asymptote, we need to determine the degree of the numerator and denominator. The degree is determined by the largest power of x in the expression. For 3x-1, the highest power is 1. It is also 1 for x+4. When the degrees are equal,
then the ratio of the leading coefficients gives the horizontal asymptote. The leading coefficient is the number in front of the x with the highest power. In the numerator it is 3; in the denominator it is 1. (Remember, x = 1x not 0x.) 3/1 = 3 so the horizontal
asymptote is y=3. You can verify this by graphing.
Let's look at g(x) = (x+7)/(x^2-4).
Find the values that make the denominator 0.
x^2-4=0 (I am going to solve by factoring.)
(x-2)(x+2)=0 => x-2=0 and x+2=0 => x = 2 and x = -2.
Neither 2 or -2 make the numerator 0. (Check by plugging in.) So the function as two vertical asymptotes at x = 2 and x = -2.
What is the degree of the numerator? 1
What is the degree of the denominator? 2
When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at y=0.
Summarizing: vertical asymptotes x=2 and x=-2; horizontal asymptote y=0.