key term - Squeeze theorem

Definition

The Squeeze Theorem states that if a function $f(x)$ is sandwiched between two other functions $g(x)$ and $h(x)$, which both approach the same limit as $x$ approaches a particular value, then $f(x)$ also approaches that limit. This theorem is useful for finding limits of functions that are difficult to evaluate directly.

5 Must Know Facts For Your Next Test

1. The Squeeze Theorem requires three functions: $f(x)$, $g(x)$, and $h(x)$ such that $g(x) \leq f(x) \leq h(x)$ for all values of $x$ in an interval around a point (except possibly at the point itself).
2. If $\lim_{x\to c} g(x) = L$ and $\lim_{x\to c} h(x) = L$, then $\lim_{x\to c} f(x) = L$.
3. The conditions must hold true in an open interval around the point of interest except possibly at the point itself.
4. It is often used when direct evaluation of a limit is challenging or impossible due to indeterminate forms.
5. This theorem can be particularly useful when dealing with trigonometric functions or oscillatory behavior.

Review Questions

Related terms

Limit:

The value that a function approaches as the input approaches some value.

Indeterminate Forms: Expressions like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ where limits cannot be directly evaluated without further analysis.

Continuity:

A property of a function if it is continuous at every point in its domain; intuitively, there are no 'jumps' or 'holes' in its graph.