Limit Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

hard

Find

\lim_{x \to 0} \frac{\sin x}{x}

Solution

1
Direct substitution gives $\frac{\sin 0}{0} = \frac{0}{0}$ , which is indeterminate.
2
This is a fundamental limit that cannot be resolved by simple algebra.
3
Using the squeeze theorem: for small positive $x$ , $\cos x \leq \frac{\sin x}{x} \leq 1$ .
4
As $x \to 0$ , $\cos x \to 1$ , so by the squeeze theorem, $\frac{\sin x}{x} \to 1$ .

Answer

\lim_{x \to 0} \frac{\sin x}{x} = 1

This is one of the most important limits in calculus. It cannot be evaluated by factoring — it requires the squeeze theorem or a geometric argument. This limit is the foundation for finding the derivative of