Asymptotes

=Asymptotes = - A line you get close to but can't touch - There are two types of asymptotes: asymptotes that are caused by discontinuities and asymptotes that relate to end behaviour

Discontinuities (vertical asymptotes)
Eg. y = 1/x-1

"Non"-Example (not all disconitnuities cause vertical asymptotes) y = __x^2 + x - 2__ <---discontinuous when x=1 x - 1

y = __(x-1) (x+2)__ <--- function has a hole at x=1 because (x-1) brackets produce the same values (they can't be cancelled out) (x-1)

End behaviour (horizontal and oblique asymptotes)
//Olique:// approaches a line that isn't horizontal Eg. let R(x)= f(x)/ g(x) where f(x) and g(x) are polynomials __**Case 1:**__ Horizontal asymptote of y=0; happens when degree of denominator is greater than the degree of the numerator R(x) = x/ x^2 + 1 __**Case 2:**__ Horizonatal asymptote y=0, a cannot = 0; when degree of numerator is the same as the denominator <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">R(x) = 3x/ x+1 <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">__**Case 3:**__ Oblique; when degree of numerator is exactly one more than denominator <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">R(x) = 3x^2 + 7x- 1/ x+1 <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">asymptote: y=3x+4 <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">__**Case 4:**__ Not oblique or horizontal asymptote; when degree of numerator is two or more than the denominator <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">R(x) = 4x^3/ x-1 <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">x-> +infinity y-> +infinity <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">x-> -infinity y-> +infinity <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">HMWK: p.262 #1-3