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Calculus Notes Mina Park

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Calculus Notes

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Calculus Notes

Limits

Limits describe how a function behaves as the input gets close to a value.

This idea lets us reason about continuity, derivatives, and local change.

Limit definition

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Limit (mathematics) Mina Park

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Limit (mathematics)

Calculus Notes

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Limit (mathematics) - Wikipedia

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en.wikipedia.org/wiki/Limit_(mathematics)

WIKIPEDIA

Limit (mathematics)

From Wikipedia, the free encyclopedia

In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.

In formulas, a limit of a function is usually written as

\lim_{x\to c} f(x) = L

and is read as «the limit of f of x as x approaches c equals L». See limit of a function for details.

History

Although implicit in the development of calculus in the 17th and 18th centuries, the modern idea of the limit goes back to Bolzano, who in 1817 introduced the basics of the epsilon-delta technique.

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Definition of the derivative Mina Park

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Definition of the derivative

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Definition of the derivative

We find the slope of the tangent line with a limit.

f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

Average rate of change
Instantaneous rate of change
Derivative

x

$x$	$f(x)=x^2$	$f'(x)=2x$
1	1	2
2	4	4
3	9	6

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paper.tex

1\documentclass{article}
2\title{The Area Under a Parabola}
3\author{A. Student}
4\begin{document}
5\maketitle
6\begin{abstract}
7Area under $y=x^2$ on $[0,1]$.
8\end{abstract}
9\section{Result}
10The area is given by
11\begin{equation}
12  
13\end{equation}
14\end{document}

PreviewPDF

The Area Under a Parabola

A. Student

Abstract

Area under y = x² on [0, 1].

1 Result

The area is given by

\int_0^1 x^2\,dx = \frac{1}{3}

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Minji

Why does

\dfrac{x^2-4}{x-2}

equal

4

x\to 2

GPT-6 Luna

The numerator

x^2-4

factors as

(x-2)(x+2)

. Cancel

(x-2)

with the denominator and only

x+2

is left, which gets close to

4

x\to 2

\lim_{x\to 2}\frac{x^2-4}{x-2}=4

Minji

Then what about

x=3

Ask anything

GPT-6 Luna

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Limits — Lesson 2

Limits

See what $f(x)=\dfrac{x^2-4}{x-2}$ gets close to as $x$ gets close to $2$ .

\lim_{x\to 2}\frac{x^2-4}{x-2}=4

$x$	$1.9$	$1.99$	$1.999$
$f(x)$	$3.9$	$3.99$	$3.999$

Cancel to get $f(x)=x+2$ , so the limit is $4$ .

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Limits — Lesson 2MM

A limit describes how a function behaves as the input gets close to a value.

\displaystyle \lim_{x\to 4}\frac{x^{2}-16}{x-4}

Mr. Leo

\displaystyle =\lim_{x\to 4}\frac{(x-4)(x+4)}{x-4}

Minji

\displaystyle =\lim_{x\to 4}(x+4)=8

MMr. LeoJust now

=\lim_{x\to 4}\frac{(x-4)(x+4)}{4-x}

Check the sign again? It should be

x-4

, not

4-x

MMinji

Thank you!

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Limit (mathematics)

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Limits — Lesson 2

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Limit (mathematics)

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Limits

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