Solve ${\lim }_{x\to 0}\frac{\sin x}{x}$

Step-by-step solution

1. Step 1 · Check by substitution
   $$\frac{\sin 0}{0}=\frac{0}{0}$$

   The indeterminate form means more work is needed.

2. Step 2 · L'Hôpital's rule
   $$\underset{x\to 0}{\lim }\frac{\cos x}{1}$$

   Differentiate numerator and denominator separately, valid for 0/0.

3. Step 3 · Evaluate
   $$\cos 0=1$$

   This limit is why the derivative of sine is cosine, in radians.