How do you find all the asymptotes for function #f(x)=[(5x+3)/(2x-3)]+1#?

1 Answer
Mar 8, 2016

Answer:

#color(green)("Horizontal asymptote at "y=5/2)#

#color(green)("Vertical asymptote at "x=3/2)#

Explanation:

The objective of this type of question is to make you think about what an equation is actually doing or saying about a situation.

Asymptotes are where the equation is getting very close to a value that is undefined or a value that does indeed exists but only as #x="infinity or zero"# is approached.

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There are two things to bear in mind with your question.

#color(brown)("Point 1.")# It contains a fraction. Thus the denominator is undefined
#color(white)(...........)#(not allowed) to become zero.

#color(brown)("Point 2.")" It will have a specific value that it tends to as "x# #color(white)(.............)" becomes enormously large"#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Consider point 1")#

#2x-3=0 => x=3/2#

#color(green)("Vertical asymptote at "x=3/2)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Consider point 2")#

As #x# becomes extremely large the equation will behave as if it is #(5/2xx x/x)+1" " =" " (5/2xx1)+1" " = " "7/2#

#color(green)("Horizontal asymptote at "y=5/2)#

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Tony B