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Limits and Derivatives

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Limits and Derivatives

The Idea of Limits

The Idea of Limits

Consider the function

The Idea of Limits

Consider the function

y

2

x

O

The Idea of Limits

Consider the function

If a function f(x) is a continuous at x0, then .

approaches to, but not equal to

The Idea of Limits

Consider the function

The Idea of Limits

Consider the function

does not exist.

A function f(x) has limit l at x0 if f(x) can be made as close to l as we please by taking x sufficiently close to (but not equal to) x0. We write

Theorems On Limits

Theorems On Limits

Theorems On Limits

Theorems On Limits

Limits at Infinity

Limits at Infinity

Consider

Generalized, if

then

Theorems of Limits at Infinity

Theorems of Limits at Infinity

Theorems of Limits at Infinity

Theorems of Limits at Infinity

Contoh 1

Contoh 2

Bila f(x) = 13

Contoh 3

Contoh 4

Contoh 5

=(6)(1)=6

Contoh 6

Contoh 7

The slope of the tangent to a curve y = f(x) with respect to x is defined as

provided that the limit exists.

Increments

The increment △x of a variable is the change in x from a fixed value x = x0 to another value x = x1.

For any function y = f(x), if the variable x is given an increment △x from x = x0, then the value of y would change to f(x0 + △x) accordingly. Hence there is a corresponding increment of y(△y) such that △y = f(x0 + △x) – f(x0).

Derivatives

The derivative of a function y = f(x) with respect to x is defined as

provided that the limit exists.

(A) Definition of Derivative.

The derivative of a function y = f(x) with respect to x is usually denoted by

The process of finding the derivative of a function is called differentiation. A function y = f(x) is said to be differentiable with respect to x at x = x0 if the derivative of the function with respect to xexists at x = x0.

The value of the derivative of y = f(x) with respect to x at x = x0 is denoted by or .

To obtain the derivative of a function by its definition is called differentiation of the function from first principles.

Jika diketahui, carilah

Jawab

Carilahkemudiancarilah

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misal

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