What is the limit when t approaches 0 of tan8t?/tan5t

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Answer 1 · 2017-02-21T08:11:26

#Lt(t->0)(tan8t)/(tan5t)=8/5#

Let us first find #Lt_(x->0)tanx/x#

#Lt_(x->0)tanx/x=Lt_(x->0)(sinx)/(xcosx)#

= #Lt_(x->0)(sinx)/x xx Lt_(x->0)1/cosx#

= #1xx1=1#

Hence #Lt_(t->0)(tan8t)/(tan5t)#

= #Lt_(t->0)((tan8t)/(8t))/((tan5t)/(5t))xx(8t)/(5t)#

= #(Lt_(8t->0)((tan8t)/(8t)))/(Lt_(5t->0)((tan5t)/(5t)))xx8/5#

= #1/1xx8/5=8/5#