Limit of tan(x)/x using known limit laws

Rewrite the expression as $\frac{\tan(x)}{x} = \frac{\sin(x)}{x}\cdot\frac{1}{\cos(x)}.$ This splits the original function into two factors whose limits are individually known or easy to compute as $x \to 0$.

By the product rule for limits, $\lim_{x\to 0}\frac{\tan(x)}{x} = \left(\lim_{x\to 0}\frac{\sin(x)}{x}\right)\left(\lim_{x\to 0}\frac{1}{\cos(x)}\right).$ The first factor equals $1$ by the standard trigonometric limit, and the second factor equals $1/\cos(0) = 1$ by direct substitution since $\cos(x)$ is continuous at $0$. Multiplying these gives $1 \cdot 1 = 1$.