+0

# Calculus

+1
190
2
+322

Find the limit

lim      sin(5x) / sin(8x)

x→0

Feb 27, 2019

#1
+106532
+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

Feb 27, 2019
#2
+23827
+1

Find the limit:

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

Apply L'Hopital's Rule and we have:

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

Feb 28, 2019