<PRE><B>Question 151671</B>
A)




{{{y=(2x+1)/(x-4))}}} Start with the given function




Looking at the numerator {{{2x+1}}}, we can see that the degree is {{{1}}} since the highest exponent of the numerator is {{{1}}}. For the denominator {{{x-4}}}, we can see that the degree is {{{1}}} since the highest exponent of the denominator is {{{1}}}.



&lt;b&gt; Horizontal Asymptote: &lt;/b&gt;

Since the degree of the numerator and the denominator are the same, we can find the horizontal asymptote using this procedure:


To find the horizontal asymptote, first we need to find the leading coefficients of the numerator and the denominator.


Looking at the numerator {{{2x+1}}}, the leading coefficient is {{{2}}}


Looking at the denominator {{{x-4}}}, the leading coefficient is {{{1}}}


So the horizontal asymptote is the ratio of the leading coefficients. In other words, simply divide {{{2}}} by {{{1}}} to get {{{(2)/(1)=2}}}



So the horizontal asymptote is {{{y=2}}}






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&lt;b&gt; Vertical Asymptote: &lt;/b&gt;

To find the vertical asymptote, just set the denominator equal to zero and solve for x


{{{x-4=0}}} Set the denominator equal to zero



{{{x=0+4}}}Add 4 to both sides



{{{x=4}}} Combine like terms on the right side



So the vertical asymptote is {{{x=4}}}



Notice if we graph {{{y=(2x+1)/(x-4)}}}, we can visually verify our answers:


{{{drawing(500,500,-10,10,-10,10,
graph(500,500,-10,10,-10,10,(2x+1)/(x-4)),
blue(line(-20,2,20,2)),
green(line(4,-20,4,20))
)}}} Graph of {{{y=(2x+1)/(x-4))}}}  with the horizontal asymptote {{{y=2}}} (blue line)  and the vertical asymptote {{{x=4}}}  (green line)




&lt;hr&gt;


B)




{{{y=(3x)/(x^2-4))}}} Start with the given function




Looking at the numerator {{{3x}}}, we can see that the degree is {{{1}}} since the highest exponent of the numerator is {{{1}}}. For the denominator {{{x^2-4}}}, we can see that the degree is {{{2}}} since the highest exponent of the denominator is {{{2}}}.



&lt;b&gt; Horizontal Asymptote: &lt;/b&gt;


Since the degree of the numerator (which is {{{1}}}) is less than the degree of the denominator (which is {{{2}}}), the horizontal asymptote is always {{{y=0}}}


So the horizontal asymptote is {{{y=0}}}




--------------------------------------------------




&lt;b&gt; Vertical Asymptote: &lt;/b&gt;

To find the vertical asymptote, just set the denominator equal to zero and solve for x


{{{x^2-4=0}}} Set the denominator equal to zero



{{{x^2=0+4}}}Add 4 to both sides



{{{x^2=4}}} Combine like terms on the right side



{{{x=2}}} or {{{x=-2}}} Take the square root of both sides

           

Notice if we graph {{{y=(3x)/(x^2-4)}}}, we can visually verify our answers:


{{{drawing(500,500,-10,10,-10,10,
graph(500,500,-10,10,-10,10,(3x)/(x^2-4)),
blue(line(-20,0,20,0)),
green(line(2,-20,2,20)),
green(line(-2,-20,-2,20))
)}}} Graph of {{{y=(3x)/(x^2-4))}}}  with the horizontal asymptote {{{y=0}}} (blue line)  and the vertical asymptotes {{{x=2}}} and {{{x=-2}}}  (green lines)
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