Find the limit. lim _(xarrow 0)(tan(9x))/(xcos(5x))=

To find the limit of the given expression, we can use the properties of limits and trigonometric functions. <br /><br />First, let's rewrite the expression:<br /><br />$\lim _{x\rightarrow 0}\frac {\tan(9x)}{x\cos(5x)}$<br /><br />We can rewrite $\tan(9x)$ as $\frac{\sin(9x)}{\cos(9x)}$ and $\cos(5x)$ remains the same. So, the expression becomes:<br /><br />$\lim _{x\rightarrow 0}\frac {\sin(9x)}{x\cos(9x)\cos(5x)}$<br /><br />Now, we can separate the limit into three parts:<br /><br />$\lim _{x\rightarrow 0}\frac {\sin(9x)}{x}\cdot\frac {1}{\cos(9x)}\cdot\frac {1}{\cos(5x)}$<br /><br />We know that $\lim _{x\rightarrow 0}\frac {\sin(9x)}{x} = 9$ and $\lim _{x\rightarrow 0}\frac {1}{\cos(9x)} = 1$ and $\lim _{x\rightarrow 0}\frac {1}{\cos(5x)} = 1$.<br /><br />Therefore, the limit of the given expression is:<br /><br />$9 \cdot 1 \cdot 1 = 9$<br /><br />So, the limit of the given expression is 9.