How do you find vertical, horizontal and oblique asymptotes for #y=1/(2-x)#?

Question

Sadie Alexander · Accepted Answer

Vertical asymptote #x=2#
Horizontal asymptote #y=0#

#color(blue)("Determine vertical asymptotes")#

Mathematically you are not allowed to divide by 0. This situation is called 'undefined'.

So at #x=2# the equation is undefined. Thus #x=2# is an asymptote.

Suppose #x>2# then #2-x < 0#

As #x# becomes increasingly closer to 2 then #2-x# becomes smaller and smaller but still negative.

In the same way, when #x<2# but approaching 2 then #2-x# becomes smaller and smaller but is still positive.
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So #1/(2-x)# becomes increasingly larger but negative or positive depending on what side of 2 we find #x# to be.

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When writing limits; when approaching a value from the positive side then for the value 2 it would be written as #color(white)()^(+)2#

Same way for approaching from the negative side it is #color(white)()^(-)2#

Thus

# y = lim_(xtocolor(white)()^(+)2)1/(2-x) = - oo " "larr ( x > 2)#

# y = lim_(xtocolor(white)()^(-)2)1/(2-x) = + oo " "larr (x<2)#

#color(green)("So "x=2 " is the vertical asymptote")#

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#color(blue)( "Determine horizontal asymptotes")#

# y = lim_(xtocolor(white)()^(+) oo)1/(2-x) = 1/(2-oo) =- 1/oo = color(white)()^(-) 0#

# y = lim_(xtocolor(white)()^(-) oo)1/(2-x) = 1/(2+oo) = +1/oo = color(white)()^(+) 0#

#color(green)("So "y=0 " is the horizontal asymptote")#

Ariana Alvarez · Answer

undefined Vertical asymptotes occur where the denominator of the rational function equals zero, so set $2 - x = 0$ and solve for $x$. For horizontal asymptotes, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $y = 0$. If the degrees are equal, divide the leading coefficients of both terms to find the horizontal asymptote. If the degree of the numerator is greater, there's no horizontal asymptote. For oblique asymptotes, if the degree of the numerator is exactly one greater than the degree of the denominator, perform polynomial long division to find the equation of the oblique asymptote. Otherwise, there is no oblique asymptote.