Question: What are the vertical and horizontal asymptote of f(x)=3x-1/12x+4
Vertical asymptotes occur when there is a division by zero. (i.e. the denominator = 0)
y = (3x - 1)/(12x + 4)
The denomijnator, (12x + 4) is zero when x = -3.
Vertical asymptote occurs at x = -3
Horizontal asymptotes are those values that the function tends to as x tends to plus or minus infinity.
y = (3x - 1)/(12x + 4)
y = (3x + 1 - 2)/{4(3x + 1)}
y = (3x + 1)/{4(3x + 1)} - 2/(4(3x + 1)}
y = 1/4 - 1/(6x + 2)
As x tends to plus or minus infinity, 1/(6x + 2) tends to zero.
So y tends to 1/4.
The horizontal asymptote is y = 1/4