Question 155662
Find the horizontal asymptote, if any, of the graph of the rational function. 
{{{g(x)=(6x^2)/(2x^2+1)}}} 
a){{{y=3}}}
b){{{y=1/3}}}
c){{{y=0}}}
d){{{no_horizontal_asymptote}}} 

<pre><font size = 4 color = "indigo"><b>
Rules for horizontal asymptotes:

1. If the largest exponent of x in the numerator is GREATER than the
largest exponent of x in the denominator, there is 
{{{no_horizontal_asymptote}}}.

2. If the largest exponent of x in the numerator is LESS than the
largest exponent of x in the denominator, the horizontal asymptote is the
x-axis, whose equation is 

{{{y = 0}}}

3. If the largest exponent of x in the numerator is EQUAL to the
largest exponent of x in the denominator, the horizontal asymptote is 
horizontal line whose equation is

{{{y = (COEFFICIENT_OF_LARGEST_POWER_OF_x_IN_THE_NUMERATOR)/(COEFFICIENT_OF_LARGEST_POWER_OF_x_IN_THE_DENOMINATOR)}}}  

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In your function:

{{{g(x)=(6x^2)/(2x^2+1)}}}

the largest exponent of x in the numerator is 2, and 
the largest exponent of x in the denominator is also 2.
They are equal so we use rule 3:

the horizontal asymptote is 
horizontal line whose equation is

{{{y = (COEFFICIENT_OF_LARGEST_EXPONENT_OF_x_IN_THE_NUMERATOR)/(COEFFICIENT_OF_LARGEST_EXPONENT_OF_x_IN_THE_DENOMINATOR)}}} 

The coefficient of the largest exponent of x in the numerator is
6, and the coefficient of the largest exponent of x in the 
denominator is 2, so the horizontal asymptote is the line
whose equation is:

{{{y=6/2}}}
{{{y=3}}}

So the correct choice is (a)

Edwin</pre>