Which is an asymptote of the graph of the function y = tan(3x/4)?
x = -4π/3, x = -2π/3, x = 3π/4, x = 3π/2

Solution:

We know:

An asymptote will be a line to a function which the function f(x) will approach under the following conditions:

\(\lim_{x\rightarrow 0}f(x) = \pm \infty\) --- (1)

\(\lim_{x\rightarrow \infty }f(x) = \pm 0\) --- (2)

The given function is :

f(x) = tan(3x/4)

We know,

tan x = sin(x)/cos(x)

The tanx function will approach ∞ as x will approach π/2, -π/2, 3π/2, -3π/2 …..and so on.

Hence the asymptotes will be pi/2, -pi/2, 3pi/2, -3pi/2 .. respectively.

The given function tan(3x/4) will approach ∞ when (3x/4) shall approach π/2, -π/2, 3π/2, -3π/2 …..

If x = -2π/3 then

tan(3x/4) = tan(3/4)(-2π/3)

= tan(-π/2) 

= sin(-π/2)/cos(-π/2)

= -1/-∞

= -∞

Hence the asymptote is x = -2π/3.

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Which is an asymptote of the graph of the function y = tan(3x/4)?

Summary:

The asymptote of the graph of the function y = tan(3x/4) will be x = -2π/3.