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 $f(x)=(3x-1)/(x+4)$ :

enter image source here

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