A limit is an idea of looking at what happens to a function as you approach particular values of x. Left-hand and right-hand limits are the idea of looking at what happens to a function as you approach a particular value of x, from a particular direction.
The limit of f(x) as x approaches the value of a from the left is written.
and the limit of f(x) as x approaches the value of a from the right is written
Definition of a Limit at a Point:
Therefore, if the left-hand limit does not equal the right-hand limit as x approaches a, then the limit as x approaches a does not exist.
Limits found Numerically and Algebraically
While almost all limits can be found graphically, as we have been discussing, it is not always practical or necessary if the function is defined algebraically.
For instance, say we are given that . If we are looking for, we can find both the left- and right-hand limits by using tables. By choosing x values that get closer and closer to x = 3 from both sides, we can analyze the behavior of f(x).
|Limit from the left|
|limit from right|
Notice that when we chose values on either side of x = 3, they were values that were very close to x = 3. It seems that as x approaches 3 from either side, the function values are approaching 6. Therefore, it seems reasonable to conclude that.
Making tables can still be as time-consuming as graphing, so we will use the following rules to algebraically evaluate limits more efficiently. Most of these rules can intuitively be verified from looking at the previously worked examples.
A function f(x) is continuous at x = a, if all of the following are true:
Continuity, this means that a function is continuous wherever the graph of the function has no holes, gaps, or jumps. A function is said to be discontinuous at x = a, if a hole, gap, or break occurs in the graph at x = a, meaning the function violates one of the three items above.
Notice that the third item in the definition of continuity and Rule 2 of the Limit Rules show that all polynomial functions are continuous for all real values of x.
Using the graph of f(x) below, find all values of x where f(x) is discontinuous and state why f(x) is discontinuous at these points, according to the definition of continuity.
x = -3, f(-3) is undefined
x = -2, f(-2) is undefined
x = 1, while the function is defined by f(1)=5 and , these are not equal and thus the third item of the definition is violated
x = 4, does not exist.
3. An important formula of limit:
4. The Differentiability
Suppose y = f (x), then its derivative with respect to x, is commonly denoted by
f ′(x) = y′ = D f (x) = Dx f (x)
The symbols and D are called differential operators. They are used to explicitly denote the differentiation of the function that follows
Basic Differentiation Formulas.
Suppose f and g are differentiable functions, c is any real number, then
Integration is the reverse process of differentiation. It is sometimes called anti-differentiation. The topic of integration can be approached in several different ways. Perhaps the simplest way of introducing it is to think of it as differentiation in reverse.
1.1 Differentiation in Reverse (Anti-Derivative)
Suppose we differentiate the function F(x) = 3x2 +7x-2. We obtain its derivative as.
This process is illustrated in Figure 1.
In this case, we can say that the derivative of F(x) = 3x2 +7x-2. is equal to 6x+7 . However, there are many other functions which also have a derivative 6x+7. Some of these are 3x2 +7x+3, 3x2 +7x,3x2 +7x-11 and so on. The reason why all of these functions have the same derivative is that the constant term disappears during differentiation. So, all of these are anti-derivatives of . Given any anti-derivative of f(x) , all others can be obtained by simply adding a different constant. In other words.
if F() is an anti-derivative of , then so too is F(x)+C for any constant and this actually describes the definition of Indefinite Integration.
1.2 INDEFINITE INTEGRATION
We call the set of all anti-derivatives of a function as the indefinite integral of the function. The indefinite integral of the function f(x) is written as
and read as "the indefinite integral of f(x) with respect to x". The function f(x) that is being integrated is called the integrand, and the variable x is called the variable of integration and the C is called the constant of integration.
1.3 Properties of the Indefinite Integral
Basically, there are three properties of anti-derivatives which been applied in order to solve the integration for any kind of functions.
1.4 Integral of Polynomial Functions
Properties of the Integral of Polynomial Functions
1.5 Integral of Exponential Functions
Formula of the Integral of Exponential Functions
1.6 Integral of Logarithmic Functions
Formula of Integral of Logarithmic Functions
1.7 DEFINITE INTEGRATION
In this section, the concept of a “definite integrals” is introduced which will link the concept of area to other important concepts such as length, volume, density, probability, and work.
Based on the Figure, the curve f(x) s nonnegative and continuous on an interval [a,b]. The area of which is under A the graph of f(x) over the interval [a,b] can be represented by the definite integral.
Note that there is no constant in definite integral, therefore definite integral is always in number. This is because the constant c is eliminated as shown below.
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