Find the horizontal and vertical asymptotes of the curve.
y = 3x+6/x−7
x=
y=
Did you mean
y = (3x+6)/(x-7) ??
Vertical asymptotes:
What happens when the denominator is zero?
For what value of x does this happen ?
horizontal :
Which terms fall by the wayside as x→∞ ?
what remains of y = (3x ...)/(x ....) ?
To find the horizontal asymptote of the curve, we need to examine the behavior of the function as x approaches positive or negative infinity.
First, let's consider the limit as x approaches positive infinity:
lim(x→∞) (3x + 6)/(x - 7)
To determine this limit, we need to compare the degrees of the numerator and denominator. In this case, the degrees are both 1. So, we divide the coefficients of the highest degree terms in the numerator and denominator.
lim(x→∞) (3x + 6)/(x - 7) = 3/1 = 3
Therefore, the horizontal asymptote of the curve is y = 3.
Next, let's find the vertical asymptote(s) of the curve. Vertical asymptotes occur when the denominator of a function becomes zero.
Set the denominator equal to zero and solve for x:
x - 7 = 0
x = 7
So, the vertical asymptote of the curve is x = 7.
To summarize:
Horizontal asymptote: y = 3
Vertical asymptote: x = 7