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
Find the limit:
\(\large{ \lim \limits_{x \to 0} ~\dfrac{\sin(5x)}{\sin(8x)} } \)
Apply L'Hopital's Rule and we have:
\(\begin{array}{|rcll|} \hline && \mathbf{\lim \limits_{x \to 0} ~\dfrac{\sin(5x)}{\sin(8x)}} \\\\ &=& \lim \limits_{x \to 0} ~\dfrac{5\cos(5x)}{8\cos(8x)} \quad | \quad \lim \limits_{x \to 0} ~ \cos(5x)= 1,\ \lim \limits_{x \to 0} ~ \cos(8x)= 1 \\\\ &=& \dfrac{ 5\cdot 1}{8\cdot 1} \\\\ &\mathbf{=}& \mathbf{\dfrac{ 5 }{8 }} \\ \hline \end{array}\)