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

enter image source here

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