I believe this should say "such that the asymptotes of the curve..."thereddevils said:Homework Statement
Find the set of values of m such that the asymtote of the curve,
[tex]y=\frac{3(m+1)x+m-2}{(m-2)x+3m}[/tex] intersect at a point above the line y=3x-2
thereddevils said:Homework Equations
The Attempt at a Solution
Vertical asymtote, x=-3m/(m-2)
horizontal asymtote, y=3(m+1)/(m-2)
i am not sure how to move on.
Mark44 said:I believe this should say "such that the asymptotes of the curve..."
The vertical and horizontal asymptotes intersect at (-3m/(m - 2), 3(m + 1)/(m - 2)). For this point to be above the corresponding point on the line y = 3x - 2, what are the conditions on m?
Mark44 said:No, it asks you to find the set of values of m such that the asymptotes of the curve...
intersect at a point above the line y=3x-2.
The two asymptotes intersect at (-3m/(m - 2), 3(m + 1)/(m - 2)). What are the coordinates of the line y = 3x - 2 when x = -3m/(m - 2)?
What are the conditions on m so that the asymptote intersection is above the point on the line at which x = -3m/(m - 2)?
Mark44 said:Set up the inequality with the y value at the point of intersection (of the asymptotes) on one side, and the y value of the line on the other. For both points, use the same x value.
An asymptote of a curve is a straight line that the curve approaches but does not intersect. It can be thought of as the "limit" of the curve as it extends infinitely in either direction.
To find the asymptotes of a curve, you can use the equation of the curve and take the limit as x approaches infinity or negative infinity. If the limit is a constant value, then that value represents the horizontal asymptote. If the limit is infinity or negative infinity, then the curve has a vertical asymptote at that value of x.
Yes, a curve can have multiple asymptotes. A curve can have both vertical and horizontal asymptotes, and it is also possible for a curve to have multiple vertical asymptotes at different values of x.
Asymptotes are important in curve problems because they help us understand the behavior of the curve as x approaches infinity or negative infinity. Asymptotes also help us identify any gaps or breaks in the curve and can provide valuable information about the overall shape of the curve.
No, not all curves have asymptotes. Some curves may intersect the horizontal or vertical line that could potentially be an asymptote, while others may not have any asymptotes at all. It depends on the equation and the behavior of the curve as x approaches infinity or negative infinity.