Q:

evaluate the limit using L'Hospitals rule

$$\lim_{x\to 0} (\csc x - \frac 1x)$$
i have tried using the L'Hopitals rule on it in 3 successive derivations and haven't been able to come to a solid conclusion. the denominator just keeps getting longer and harder to differentiate while the numerator keeps switching between sinx and cosx. i believe there might be a simpler answer but unfortunately i can't seem to be able to get there.

A:

The first step should be to find a common denominator:
$$\lim_{x \to 0} \frac{x-\sin{x}}{x\sin{x}}$$
Then apply L'Hospital's rule:
$$\lim_{x \to 0} \frac{1-\cos{x}}{\sin{x}+x\cos{x}}$$
Apply L'Hospital's rule again:
$$\lim_{x \to 0} \frac{\sin{x}}{\cos{x}+\cos{x}-x\sin{x}}=0$$

