Squeeze Theorem

The Squeeze Theorem

The Squeeze Theorem is a powerful technique for evaluating limits that resist standard algebraic methods. It works by trapping a difficult function between two simpler ones.

The Core Idea

If $g(x) \le f(x) \le h(x)$ for all $x$ near $a$, and $\lim_{x\to a} g(x) = \lim_{x\to a} h(x) = L$, then $\lim_{x\to a} f(x) = L$.

Intuitively, as $x$ approaches $a$, the gap between $g(x)$ and $h(x)$ shrinks to zero. Since $f(x)$ is always trapped inside that shrinking gap, it must converge to $L$ as well — even without a direct formula for $f$'s limit.

Essential Trig Limits

The theorem's most important application proves two cornerstone results:

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

Direct substitution yields $\frac{0}{0}$, which is indeterminate. Using geometric arguments (comparing areas of triangles and circular sectors), we can show that for $x$ near $0$:

$\cos x \le \frac{\sin x}{x} \le 1$

Since both bounds converge to $1$, the Squeeze Theorem forces the middle expression to $1$ as well.

A companion result, derived similarly, is:

$\lim_{x\to 0} \frac{1-\cos x}{x} = 0$

Practical Examples

Example 1: Find $\lim_{x\to 0} \frac{\sin(3x)}{x}$.

Rewrite as $3 \cdot \frac{\sin(3x)}{3x}$. With $u = 3x$, this becomes $3 \cdot \lim_{u\to 0}\frac{\sin u}{u} = 3 \cdot 1 = 3$.

Example 2: Find $\lim_{x\to 0} x^2 \sin\!\left(\frac{1}{x}\right)$.

Here $\sin(1/x)$ oscillates wildly with no limit. However, $-1 \le \sin(1/x) \le 1$, so:

$-x^2 \le x^2\sin\!\left(\frac{1}{x}\right) \le x^2$

Since both bounds approach $0$, the Squeeze Theorem gives $\lim_{x\to 0} x^2\sin(1/x) = 0$.

Key Takeaway

The Squeeze Theorem is indispensable for oscillating functions and trigonometric limits. Find bounds that trap your function, verify the bounds share a limit, and conclude the trapped function has the same limit.