Question 132442
I'm not sure as to what you mean by "Discuss completely (as in the textbook)", but I'm assuming that you want to find the asymptotes right?






{{{y=(x^2-4)/(x))}}} Start with the given function




Looking at the numerator {{{x^2-4}}}, we can see that the degree is {{{2}}} since the highest exponent of the numerator is {{{2}}}. For the denominator {{{x}}}, we can see that the degree is {{{1}}} since the highest exponent of the denominator is {{{1}}}.



<b> Oblique Asymptote: </b>


Since the degree of the numerator (which is {{{2}}}) is greater than the degree of the denominator (which is {{{1}}}), there is no horizontal asymptote. In this case, there's an oblique asymptote


To find the oblique asymptote, simply use polynomial division to find it. The quotient of {{{(x^2-4)/(x))}}} is the equation of the oblique asymptote



<pre>

  __x_________
x | x^2 - 4
    x^2
   -----
        - 4

</pre>


note: in this case, we don't need to worry about the remainder


Since the quotient is {{{x}}}, this means that the oblique asymptote is {{{y=x}}}





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<b> Vertical Asymptote: </b>

To find the vertical asymptote, just set the denominator equal to zero and solve for x


{{{x=0}}} Set the denominator equal to zero



So the vertical asymptote is {{{x=0}}}



Notice if we graph {{{y=(x^2-4)/(x)}}}, we can visually verify our answers:


{{{drawing(500,500,-10,10,-10,10,
graph(500,500,-10,10,-10,10,(x^2-4)/(x),100,x),
green(line(0,-20,0,20))
)}}} Graph of {{{y=(x^2-4)/(x))}}}  with the oblique asymptote {{{y=x}}} (blue line)  and the vertical asymptote {{{x=0}}}  (green line)