v) $$ \lim _{x \rightarrow 0}(1-\sin 2 x)^{\frac{3}{x}} $$

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Calculate the limit $$ \lim_{x ightarrow 0} (1-\sin(2x))^{\frac{3}{x}} $$.
Evaluate the limit by following steps:
- step0: Evaluate using L'Hopital's rule:
  $$\lim _{xightarrow 0}\left(\left(1-\sin\left(2xight)ight)^{\frac{3}{x}}ight)$$
- step1: Transform the expression:
  $$\lim _{xightarrow 0}\left(e^{\frac{3}{x}	imes \ln{\left(1-\sin\left(2xight)ight)}}ight)$$
- step2: Transform the expression:
  $$e^{\lim _{xightarrow 0}\left(\frac{3}{x}	imes \ln{\left(1-\sin\left(2xight)ight)}ight)}$$
- step3: Use the L'Hopital's rule:
  $$e^{-6}$$
- step4: Express with a positive exponent:
  $$\frac{1}{e^{6}}$$
The limit of $$ \lim _{x ightarrow 0}(1-\sin 2 x)^{\frac{3}{x}} $$ is $$ \frac{1}{e^{6}} $$.
The limit is $$ \frac{1}{e^{6}} $$.

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