Tanx/sin2x? limit x->0

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Answer 1 · 2018-06-22T15:33:34

#lim_(x->0) tanx/sin(2x) = 1/2#

Consider the fundamental trigonometric limit:

#lim_(x->0) sinx/x =1#

and note that also:

#lim_(x->0) tanx/x =lim_(x->0) 1/cosx sinx/x = 1#

Then:

#lim_(x->0) tanx/sin(2x) = lim_(x->0) 1/2 tanx/x (2x)/sin(2x)#

#lim_(x->0) tanx/sin(2x) = 1/2 lim_(x->0) tanx/x lim_(x->0)(2x)/sin(2x)#

#lim_(x->0) tanx/sin(2x) = 1/2 *1*1 = 1/2#

Answer 2 · 2018-06-22T15:35:01

#lim_(x->0) tanx/(sin2x)=1/2#

Let #L = lim_(x->0) tanx/(sin2x)#.

#L = lim_(x->0) color(red)(sinx/cosx)/color(blue)(2sinxcosx)#

#L = lim_(x->0) 1/(2cos^2x)=1/(2cos^2 0)=1/2#