Show that limit of log(1+2x)/sin3x=2/3 as x approaches to 0?

Question

Show that limit of #log_e(1+2x)/sin(3x)=2/3# as x approaches to 0?

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Answer 1 · 2017-12-08T10:50:16

# 2/3.#

Recall that, #lim_(x to 0)log_e(1+x)^(1/x)=e, &, lim_(x to 0)sinx/x=1.#

#"Now, the Reqd. Lim. L="lim_(x to 0)log_e(1+2x)/sin(3x),#

#=lim{(2x)/(2x)log_e(1+2x)}/{(sin(3x)/(3x))3x,#

#=lim{(2x)/(3x)}{1/(2x)log_e(1+2x)}/(sin(3x)/(3x)),#

#=2/3lim_((2x) to 0){log_e(1+2x)^(1/(2x))}-:lim_((3x) to 0)(sin(3x)/(3x)),#

#=2/3(log_ee-:1).#

# rArr L=2/3.#

Enjoy Maths.!