How do I determine the end behavior of the graph, f(x)=(3x-3)/(4x+5), in limit notation?

1 Answer
Apr 18, 2018

See below.

Explanation:

The end behaviour is what is happening as x->+-oo

lim_(x->oo)(3x-3)/(4x+5)

(3x-3)/(4x+5)

Divide by x:

((3x)/x-3/x)/((4x)/x+5/x)=(3-3/x)/(4+5/x)

lim_(x->oo)(3x-3)/(4x+5)=>(3-3/x)/(4+5/x)->(3-0)/(4+0)=3/4

As before:

lim_(x->-oo)(3x-3)/(4x+5)=>(3-3/x)/(4+5/x)->(3-0)/(4+0)=3/4

This shows that the line y=3/4 is a horizontal asymptote.

So as x->+-oo the function tends to 3/4.

The graph of f(x) confirms this:

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