How do you evaluate the limit #tan(4x)/x# as x approaches #0#?

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Answer 1 · 2016-10-15T13:58:01

The limit equals #4#.

Rewrite in sine and cosine using the identity #tanx = sinx/cosx#.

#=lim_(x -> 0)(sin(4x)/cos(4x))/x#

#=lim_(x->0) sin(4x)/(xcos(4x))#

Rewrite so that that one expression is #sin(4x)/x#.

#=lim_(x-> 0) sin(4x)/x xx 1/cos(4x)#

Use the well know limit that #lim_(x ->0) sinx/x = 1# to deduce the fact that #lim_(x -> 0) sin(4x)/x = 4#.

#=4 xx 1/cos(0)#

#=4 xx 1#

#= 4#

Hopefully this helps!