How do I prove that this limit is equal to 1 in a formal way ? (without using l'hospital rule)

Question

#limx->0^+ ln(sinx)/ln(x)=1#

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Answer 1 · 2017-02-03T19:48:28

I don't have a formal proof worked out yet, but

I would try using the fact that #lim_(xrarr0) sinx/x = 1#

This tells us that for #x# very close to #0#,

#x ~~ sinx#, so we should be able to conclude that

#lnx ~~ ln(sinx)# for #x# near #0#.