This title slide is headed "History of LIMIT." Its subtitle reads "Exploring the Evolution of Limits in Mathematics."

History of LIMIT

Exploring the Evolution of Limits in Mathematics

Source: Zakariya Mochtaq Student ID: 259J82

--- Speaker Notes: Exploring the evolution of limits in math. (Print this page first!)

This section header slide, titled "History of the Limit," introduces "Ancient Origins" as section 02. It highlights roots in Greek mathematics, specifically Eudoxus' method of exhaustion (400 BC) for approximating areas without infinity.

History of the Limit

02

Ancient Origins

Roots in Greek mathematics: Eudoxus' method of exhaustion (400 BC) approximated areas without infinity.

--- Speaker Notes: Presenter: Zakariya Mochtaq, ID: 259J82. This section explores the ancient origins of limits in Greek mathematics.

This timeline slide outlines key milestones in the development of calculus, starting with Eudoxus' method of exhaustion in 400 BC for approximating areas and volumes. It continues with Cavalieri's indivisibles in the 1630s, Newton and Leibniz's independent invention in the 1660s, and Cauchy's rigorous limit definition in 1821.

Key Milestones

400 BC: Eudoxus Method of Exhaustion Greek mathematician develops technique to approximate areas and volumes precisely. 1630s: Cavalieri Introduces Indivisibles Italian mathematician uses indivisibles as precursor to integral calculus. 1660s: Newton and Leibniz Invent Calculus Develop calculus independently with intuitive limits for derivatives and integrals. 1821: Cauchy Rigorizes Limit Definition Provides first rigorous mathematical definition of limits in analysis.

Source: History of Limits in Calculus

--- Speaker Notes: Prepared by Zakariya Mochtaq, ID: 259J82. For PowerPoint on history of LIMIT and print submission.

In the 17th century, Isaac Newton developed fluxions while Gottfried Leibniz introduced infinitesimal methods, foundational to calculus. Implicit limits enabled derivatives and integrals, revolutionizing physics and mathematics.

17th Century Breakthrough

• Isaac Newton developed fluxions for calculus.

• Gottfried Leibniz introduced infinitesimal methods.

• Implicit limits enabled derivatives and integrals.

• Revolutionized physics and mathematics.

The slide, titled "Cauchy's Insight," features a quote from pioneering mathematician Augustin-Louis Cauchy. The quote states: "Limit is the fundamental notion upon which modern analysis is based."

Cauchy's Insight

> Limit is the fundamental notion upon which modern analysis is based.

— Augustin-Louis Cauchy, Pioneering Mathematician

Source: 1821

--- Speaker Notes: Formalized limit process in the history of limits.

The slide covers Karl Weierstrass's 1850s rigorous ε-δ definition of limits. It states ∀ε>0 ∃δ>0: |x-a|<δ ⇒ |f(x)-L|<ε, calling it the foundation of real analysis.

Weierstrass & Epsilon-Delta

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• 1850s: Rigorous ε-δ limit definition

• ∀ε>0 ∃δ>0: |x-a|<δ ⇒ |f(x)-L|<ε

• Foundation of real analysis

Source: Wikipedia

The "Impact Statistics" slide highlights three key metrics on limits in calculus. Limits enable 90% of calculus, support 100M+ engineering calculations yearly worldwide, and form the core basis for AI optimization algorithms.

Impact Statistics

• 90%: Calculus Enabled

Limits enable 90% of calculus

• 100M+: Engineering Calcs

Used yearly worldwide

• Core: AI Optimization

Basis for algorithms

--- Speaker Notes: Student: Zakariya Mochtaq, ID: 259J82. PowerPoint slide on history of LIMIT for print submission.

The conclusion slide highlights how the concept of LIMIT evolved from ancient approximations to rigorous proofs, transforming mathematics. It credits presenter Zakariya Mochtaq (ID: 259J82) and ends with "Thank you! Questions?"

Conclusion

From ancient approximations to rigorous proofs, LIMIT transformed math.

Zakariya Mochtaq ID: 259J82

Thank you! Questions?

--- Speaker Notes: History of LIMIT presentation. Zakariya Mochtaq, ID: 259J82. Print paper: include name, class, ID on page 1.