How do you find the limit of #(Tan4x)^x# as x approaches 0?

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Answer 1 · 2017-09-19T00:14:34

# lim_(x rarr 0) (tan4x)^x =1 #

We seek:

# L = lim_(x rarr 0) (tan4x)^x #

The logarithmic function is monotonically increasing so we have:

# ln L = ln{lim_(x rarr 0) (tan4x)^x }#
# \ \ \ \ \ \ = lim_(x rarr 0) ln{(tan4x)^x }#
# \ \ \ \ \ \ = lim_(x rarr 0) xln(tan4x)#

Note that both #tanx rarr 0# and #x rarr 0# uniformly as #x rarr 0#, and so:

# ln L = 0 xx 0#
# \ \ \ \ \ \ = 0 #

And so:

# L = e^ 0 =1 #