How do you find the vertical, horizontal and slant asymptotes of: #y = (x^2 + 7)/( 5x - 4x^2)#?

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Answer 1 · 2017-03-09T12:56:09

The vertical asymptotes are #x=0# and #x=5/4#
The horizontal asymptote is #y=-1/4#
No slant asymptote

We cannot divide by #0#, so the denominator cannot #=0#

So,

#5x-4x^2!=0#

#x(5-4x)!=0#

Therefore, #x!=0# and #x!=5/4#

The vertical asymptotes are #x=0# and #x=5/4#

The degree of the the numerator #=# the degree of the denominator, there is no slant asymptote

#lim_(x->+-oo)y=lim_(x->+-oo)-x^2/(4x^2)=-1/4#

The horizontal asymptote is #y=-1/4#

graph{(y-(x^2+7)/(5x-4x^2))(y+1/4)(y-1000(x-5/4))=0 [-13.42, 14.3, -4.98, 8.89]}