$\lim _{x \rightarrow 0} \frac{\tan ^{-1}(9 x)-9 x \cos (9 x)-\frac{243}{2} x^{3}}{x^{5}}=$

Full solution

\lim _{x \rightarrow 0} \frac{\tan ^{-1}(9 x)-9 x \cos (9 x)-\frac{243}{2} x^{3}}{x^{5}}=

Identify Expression and Limit: Identify the expression to simplify and set up the limit problem. $\newline$ $\lim_{x \to 0}\frac{\tan^{-1}(9x) - 9x \cos(9x) - \frac{243}{2}x^3}{x^5}$
Apply Taylor Series Expansions: Apply Taylor series expansions for $\tan^{-1}(9x)$ and $9x \cos(9x)$ around $x = 0$ . $\newline$ $\tan^{-1}(9x) \approx 9x - \frac{(9x)^3}{3} + \frac{(9x)^5}{5}$ $\newline$ $9x \cos(9x) \approx 9x - \frac{(9x)^3}{2}$
Substitute Expansions into Limit: Substitute the expansions into the limit expression. $\newline$ $\lim_{x \to 0}\frac{9x - \frac{729x^3}{3} + \frac{59049x^5}{5} - (9x - \frac{729x^3}{2}) - \frac{243}{2}x^3}{x^5}$
Simplify Numerator: Simplify the expression in the numerator. $\newline$ $\lim_{x \to 0}\frac{9x - 9x + \frac{729x^3}{2} - \frac{729x^3}{3} - \frac{243}{2}x^3 + \frac{59049x^5}{5}}{x^5}$ $\newline$ $\lim_{x \to 0}\frac{\frac{243x^3}{6} + \frac{59049x^5}{5}}{x^5}$
Further Simplify and Reduce: Further simplify by canceling out terms and reducing. $\newline$ $\lim_{x \to 0}\frac{\frac{243x^3}{6} + \frac{59049x^5}{5}}{x^5} = \lim_{x \to 0}\left(\frac{243}{6x^2} + \frac{59049}{5}\right)$