How do you find the limit of #tan(pi*x)/x# as x approaches 0?

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Answer 1 · 2016-08-07T18:33:11

#pi#

#lim_{x to 0} tan(pi*x)/x#

#= lim_{x to 0} 1/(cos pi x)(sin pi*x )/x#

and because #cos pi x# is continuous through the limit and is 1 at the limit

#= lim_{x to 0} (sin pi*x )/x#

#= lim_{x to 0} pi (sin pi*x )/(pi x)#

with #y = pi x#
#=pi lim_{y to 0} (sin y )/(y) #

#= pi#

as # lim_{y to 0} (sin y )/(y) = 1# is a well known result