Question

lim x tendendo 0 de tg 2x/3x

Answer 1

Lim  (tg (2x))/3x
x->0

Lim   (sen 2x /cos 2x) /3x
x-->0

Lim   (sen 2x /3x*cos 2x)
x-->0

Lim   (2/2) *(sen 2x /3x*cos 2x)
x-->0

Lim   (2/3) *(sen 2x /2x) * (1/cos 2x)
x-->0


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Vou usar a seguinte propriedade do Limite:
O limite do produto e o produto dos limites
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Lim   (2/3) *(sen 2x /2x) * Lim  (1/cos 2x)
x-->0                                x--> 0


(2/3) *Lim  (sen 2x /2x) * Lim  (1/cos 2x)          
x-->0


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Também vou usar o limite fundamental

Lim       sen(y) =1              ...sempre será igual a 1
y-->0     -------                 
                y

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(2/3) *Lim  (sen 2x /2x) * Lim  (1/cos 2x) 
         x-->0                     x-->0

(2/3) *   1 *  (1/cos 2*0)   =(2/3)*1 * 1/cos 0 

 =(2/3)*1*1/1 =2/3  é a resposta