Limits and Derivatives

Limits and DerivativesThe Idea of LimitsConsider the function The Idea of Limits

x1.91.991.9991.999922.00012.0012.012.1f(x)Consider the function The Idea of Limits

x1.91.991.9991.999922.00012.0012.012.1f(x)3.93.993.9993.9999un-defined4.00014.0014.014.1

Consider the function The Idea of Limits

x1.91.991.9991.999922.00012.0012.012.1g(x)3.93.993.9993.999944.00014.0014.014.1

xyO2If a function f(x) is a continuous at x0, then .

approaches to, but not equal to Consider the function The Idea of Limits

x-4-3-2-101234g(x)Consider the function The Idea of Limits

x-4-3-2-101234h(x)-1-1-1-1un-defined1234

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 InfinityLimits at InfinityConsider

Generalized, if

then

Theorems of Limits at Infinity

Theorems of Limits at Infinity

Theorems of Limits at Infinity

Theorems of Limits at InfinityContoh - contohThe Slope of the Tangent to a Curve

The Slope of the Tangent to a Curve

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

provided that the limit exists.IncrementsThe 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(A) Definition of Derivative.The derivative of a function y = f(x) with respect to x is defined as

provided that the limit exists.

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 x exists 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.Contoh SoalRumus-Rumus Diferensial

Contoh - contohSoal Latihan