# What is the limit as x approaches 0 of #(2x)/tan(3x)#?

# lim_(x rarr 0) (2x)/tan(3x) = 2/3#

The limit:

And so applying L'Hôpital's rule we get:

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To determine the limit graphically, se the graph below.

graph{2x/tan(3x) [-6.17, 6.316, -2.69, 3.547]}

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Seeing that plugging in gives an indeterminate form, we need to use L'Hospital's Rule.

Note that I moved the 2 on top of the limit outside of the limit.

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# lim_(x rarr 0) (2x)/tan(3x) = 2/3#

Another approach that doesn't rely on using L'Hôpital's rule

We can write the limit as:

And so we can write a series expansion for the limit:

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The limit as x approaches 0 of (2x)/tan(3x) is 2/3.

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