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+0  
 
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avatar+301 

Find the limit

 

 lim      sin(5x) / sin(8x)

x→0     

 Feb 27, 2019
 #1
avatar+99582 
+2

lim               sin 5x

x →0           _____   

                    sin 8x

 

Note that we can write

 

 

lim               sin (5x)

x → 0        _________   

                        x

            _______________              and we   can  also write

                   sin (8x)

                  ______

                       x 

 

 

lim               5sin (5x)

x →0           _______

                      5x

              _____________

                  8sin (8x)

                  ________

                    8x

 

Note that  using a limit property   .....if we let 5x , 8x =  θ  

 

lim                 sin  θ

 θ →0          _______  =   1

                         θ

 

So we have

 

5(1)           5 

___   =     ___  

8(1)           8

 

 

cool cool cool

 Feb 27, 2019
 #2
avatar+21978 
+1

Find the limit:

\(\large{ \lim \limits_{x \to 0} ~\dfrac{\sin(5x)}{\sin(8x)} } \)

 

Apply L'Hopital's Rule and we have:

\(\begin{array}{|rcll|} \hline && \mathbf{\lim \limits_{x \to 0} ~\dfrac{\sin(5x)}{\sin(8x)}} \\\\ &=& \lim \limits_{x \to 0} ~\dfrac{5\cos(5x)}{8\cos(8x)} \quad | \quad \lim \limits_{x \to 0} ~ \cos(5x)= 1,\ \lim \limits_{x \to 0} ~ \cos(8x)= 1 \\\\ &=& \dfrac{ 5\cdot 1}{8\cdot 1} \\\\ &\mathbf{=}& \mathbf{\dfrac{ 5 }{8 }} \\ \hline \end{array}\)

 

 

laugh

 Feb 28, 2019

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