Limits

Limits formalize the idea of a function approaching a value as its input gets close to a specific point or grows without bound. This topic covers evaluating basic limits, applying L'Hopital's rule and the squeeze theorem, analyzing limits at infinity, testing continuity, and constructing epsilon-delta proofs.

Limits are the theoretical foundation of calculus: both derivatives and integrals are defined as limits. Understanding them is essential for rigorous work in analysis and applied mathematics.

Practice Tips

• 1Always try direct substitution first; if you get 0/0 or infinity/infinity, then apply L'Hopital's rule or algebraic simplification.
• 2For limits at infinity of rational functions, divide every term by the highest power of x in the denominator to see which terms vanish.
• 3In epsilon-delta proofs, work backwards: start with the inequality you need to prove and figure out what delta must be in terms of epsilon.