Question

Solve: $\underset{x\to 0}{lim}(\frac{1}{x^2}-{\cot }^{2}x)$.

Answer 1

$\underset{x\to 0}{lt}(\frac{1}{x^2}-{\cot }^{2}x)$

$\Rightarrow \underset{x\to 0}{lt}(\frac{1}{{\sin }^{2}x}-\frac{{\cos }^{2}x}{{\sin }^{2}x})$

$\because \underset{x\to 0}{lt}\sin x=x$

$\Rightarrow \underset{x\to 0}{lt}\frac{1-{\cos }^{2}x}{{\sin }^{2}x}=\underset{x\to 0}{lt}\frac{\text{/}{\sin }^{2}x}{\text{/}{\sin }^{2}x}=1$.