#lim_(theta rarr 0) (sinmtheta)/(sinntheta)#? By not using l'hospital rule

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Answer 1 · 2017-08-27T13:06:52

# m/n#.

We know #lim_(x to 0) sinx/x=1..............(ast),# we have,

#"The Reqd. Lim.="lim_(theta to 0) sin(mtheta)/sin(ntheta),#

#=lim_(theta to 0){sin(mtheta)/(mtheta)m}/{sin(ntheta)/(ntheta)n}.....(1).#

We note that, by #(ast),# both the limits - in Nr., and, Dr. - exist :

#lim_(theta to 0) sin(mtheta)/(mtheta)=1............(2),#

# lim_(theta to 0) sin(ntheta)/(ntheta)=1..................(3).#

Utilising #(2), and, (3)" in "(1),# we get,

#"The Reqd. Lim.="m/n.#