Operations on Limits and Comparison Theorems

This page delves into operations performed on limits and introduces important comparison theorems, including the Squeeze Theorem.

Operations on Limits

The guide outlines how to handle limits of sums, products, and quotients of functions. It provides rules for various scenarios, including cases involving infinity.

Example: For the sum of limits: lim (f+g)(x) = lim f(x) + lim g(x) x→a x→a x→a

Highlight: Special attention is given to cases involving infinity, such as ∞ - ∞, which is indeterminate.

Comparison Theorems

The page introduces comparison theorems, which are crucial for evaluating limits in complex situations.

Definition: The Squeeze Theorem (Théorème des gendarmes) states that if g(x) ≤ f(x) ≤ h(x) and lim g(x) = lim h(x) = L, then lim f(x) = L. x→a x→a x→a

This theorem is particularly useful when direct computation of a limit is difficult or impossible.

Vocabulary: "Théorème des gendarmes" is the French term for the Squeeze Theorem, also known as the Sandwich Theorem in English.

The page concludes with a table summarizing various limit scenarios, providing a quick reference for students tackling limit problems.

This comprehensive coverage of limit operations and comparison theorems equips students with powerful tools for solving complex limit problems in calculus and analysis.