Calculating $\lim_{x\to0} \frac{tan^4(2x)}{4x^4}$

Question

I’m trying to solve the following limit $$\lim_{x\to0} \frac{\tan^4(2x)}{4x^4}$$ I have made this process. $$\lim_{x\to0} \frac{\frac{1}{2}\tan^4(2x)}{\frac{1}{2}4x^4}$$ then I use the identity $$ \lim_{x\to 0} \frac{\tan(x)}{x}=1$$ and the answer I get is 1, but the answer in the book is 4, so I don’t know what to do to get that answer

Answer 1

write it as $$\frac{(\tan(2x))^4}{(2x)^4}\cdot \frac{16}{4}$$

Answer 2

If we set $y=2x$ this is $$\lim_{y\to0}4\frac{\tan^4y}{y^4}=4\lim_{y\to0}\left( \frac{\tan y}{y}\right)^4.$$

Answer 3

$$\lim_{x\to 0} \frac{\tan^4(2x)}{2^{2}x^4}=2^{2}\lim_{x\to 0} \frac{\tan^4(2x)}{2^{4}x^4}=4.$$