How do I find the vertical and horizontal asymptotes of the function #f(x)=(3x-1)/(x+4)#?

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Answer 1 · 2018-04-03T15:56:31

See below.

#f(x)=(3x-1)/(x+4)#

Vertical asymptotes occur where the function is undefined.

This happens when #x=-4#

So the line #color(blue)(x=-4)# is a vertical asymptote.

We next examine the end behaviour as #x->-+-oo#

Starting with:

#(3x-1)/(x+4)#

Divide by #x#:

#((3x)/x-1/x)/(x/x+4/x)#

Cancelling:

#((3cancel(x))/cancel(x)-1/x)/(cancel(x/x)+4/x)=(3-1/x)/(1+4/x)#

as #x->oo# , #(3-1/x)/(1+4/x)->(3-0)/(1+0)=3#

as #x->-oo# , #(3-1/x)/(1+4/x)->(3-0)/(1+0)=3#

This show that the line #color(blue)(y=3)# is a horizontal asymptote.

These findings are confirmed by the graph of #f(x)=(3x-1)/(x+4)#:

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