Finding $\displaystyle \lim_{x\rightarrow 0}\frac{x^{6000}-(\sin x)^{6000}}{x^{6002}}$
Try: I have tried using series expansion of $\displaystyle \sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots $
So $$\lim_{x\rightarrow 0}\frac{x^{6000}-\bigg(x-\frac{x^3}{3!}+\cdots \bigg)^{6000}}{x^{6002}}$$
$$\lim_{x\rightarrow 0}\frac{x^{6000}-x^{6000}\bigg[1-\frac{x^2}{6}+\cdots \bigg]^{6000}}{x^{6002}} = 1000$$
Could some help me how to solve it without Series expansion or D L Hopital Rule
Thanks