How do you find all the asymptotes for function #y=(x^2-4)/(x)#?

1 Answer

Answer:

The vertical asymptote is #x=0# and oblique asymptote
#y=x#

Explanation:

A line #x=a# is a vertical asymptote of a function f(x) if

#lim_(x->a)f(x)=+-oo#

A line #y = b# is a horizontal asymptote of a function f(x) if

#lim_(x->+-oo)=b#

An oblique asymptote for a function f(x) has the formula

#y=cx+d#

where

#c=lim_(x->oo)f(x)/x# and #d=lim_(x->oo)(f(x)-cx)#

Hence we have that

#f(x)->+-oo , x->0#

and #c=limf(x)/x=lim(x^2-4)/x^2=1#

#d=lim((x^2-4)/x-x)=0#

Hence #x=0# and #y=x#