I'm having trouble finding
$$
\lim\_{x\to0}\biggl(\frac{1}{x}-\frac{1}{\tan x}\biggr)
$$

As $x$ approaches zero, the function is in indeterminate form, so I used L'Hôpital's rule and took the derivative: 
$$-\frac{1}{x^2}-\frac{\sec^2x}{\tan^2x}
$$

This is still in indeterminate form and I still can't take the limit as x approaches zero, so I took the derivative again and still got it in indeterminate form. Is there something that I'm doing wrong either algebraically or with respect to differentiation?