11.10: The Squeeze Theorem

Certain mathematical functions exhibit unpredictable or highly variable behavior near specific input values, making direct evaluation of their limits challenging. This complexity may arise from rapid oscillations or irregular patterns that obscure the function’s trend. In such cases, the Squeeze Theorem offers a reliable method for determining limits.

According to the Squeeze Theorem, if a function is confined between two other functions near a particular point, and both outer functions approach the same limit as the input nears that point, then the function trapped between them must also approach that limit. The converging behavior of the boundary functions effectively narrows the range in which the intermediate function can exist, forcing it toward the same limiting value.

The key condition for using the theorem is that the function in question must always remain greater than or equal to a lower bound and less than or equal to an upper bound near the point under consideration. If both bounds tend toward the same value, the intermediate function is determined to share this limiting behavior regardless of its internal complexity. This principle is especially useful when the function is not easily approachable through conventional techniques due to its erratic behavior.

Some functions are bounded between two other functions, which determine their limiting value.

The Squeeze Theorem applies when a function remains bounded between two other functions over an interval, usually denoted by I, near a point, and both bounding functions approach the same value. Within that interval, the function constrained between them must also share that limit.

This creates a narrowing pathway, keeping the inner curve trapped between the other two functions near a specific point.

Consider the function x² times the cosine of 20πx.

As x approaches zero, the function converges to zero.

Since cosine is always bounded between -1 and 1, multiplying by x² makes the product oscillate within the envelopes –x² and x², regardless of its frequency.

As a result, the entire function remains between two bounding functions: –x² and x². 

Both bounding functions approach zero as x approaches zero, so the middle function also approaches zero, as defined by the squeeze theorem.

This theorem also appears in engineering software, where stress estimates are bounded between upper and lower limits. With each iteration, the bounds tighten and the estimate converges.