Solve lim x→0 of sin(x)/x

Step-by-step solution

1. Direct substitution gives $\frac{\sin 0}{0} = \frac{0}{0}$ — indeterminate form.

2. Apply L'Hôpital's rule: differentiate numerator and denominator separately.

3. Numerator derivative: $\frac{d}{dx}\sin x = \cos x$.

4. Denominator derivative: $\frac{d}{dx}x = 1$.

5. Limit becomes $\lim_{x \to 0} \frac{\cos x}{1} = \cos 0 = 1$.

6. This limit is so important it has its own name: the fundamental trig limit, foundational to all of calculus.