How do you tell if it's a vertical asymptote function or a horizontal asymptote function?

2 Answers
Oct 12, 2015

Answer:

See explanation

Explanation:

To see if a function has vertical asymptote you have to find values of #x# which are not in the domain, but their surrounding is. For example if #f(x)=1/x#, then #x=0# is a vertical asymptote. To ensure that such point is an asymptote you have to calculate left and right side limits:

#lim_{x->0^+}1/x=+oo#

#lim_{x->0^-}1/x=-oo#

graph{(1/x) [-10, 10, -5, 5]}

To find if a function has horizontal (or oblique) asymptotes you have to calculate limits:

#a= lim_{x->-oo} f(x)/x# and #b=lim_{x->-oo} f(x)-ax#

If both limits are finite, then function #f(x)=ax+b# is an oblique (or horizontal for #a=0#) asymptote for #f(x)#.
For example function #f(x)=x+1/x# has an oblique asymptote #f(x)=x#

graph{(y-x-1/x)(y-x)=0 [-10, 10, -5, 5]}

Oct 14, 2015

Answer:

See the explanation.

Explanation:

Vertical

#x=h# is a vertical asymptote if #f(x)# increases or decreases without bound (#f(x)rarroo" or "-oo#) as #x# approaches #h# from at least one side.

Roughly: A vertical asymptote occurs if #y# goes to #+-# infinity as #x# is approaching some finite number

Horizontal

Roughly: A horizontal asymptote occurs if #y# goes to some finite number as #x# is goes to #+-# infinity.

#y=k# is a horizontal asymptote if #f(x) rarr k# as #x# increases or decreases without bound (#x rarr oo" or "-oo#)