How do you find the vertical, horizontal or slant asymptotes for #f(x)=(2x)/sqrt (9x^2-4)#?

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Answer 1 · 2016-07-08T17:53:51

Vertical assymptotes at #x=pm 2/3#
Horizontal assymptotes at #y=pm 2/3#

Assymptotes appear for large #abs(x)# values

For large #abs(x)# values we have

#sqrt(9 x^2-4) approx 3 abs(x)# so for large #abs(x)#

#f(x)=(2x)/sqrt (9x^2-4) approx (2 x)/(3 abs(x)) = 2/3 sign(x)#

Also we have vertical assymptotes at #x = pm2/3#

Attached a figure

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