Continuity Theorems.
The basic theorems based on continuity are given below: If the functions f(x) and g(x) are continuous at x=a then, 1> is continuous at x=a. 2> is continuous at x=a. 3> is continuous at x=a , if g(x) is not equal to 0. 4> is continuous at x=a if f(x) is greater than 0 , or is a positive number when “n” is...
read moreContinuity of a function(continuous and discontinuous functions).
A function “f” in interval [a,b] is said to be a continuous function when the Graph drawn for f(x) is a smooth line or curve without any break in it. Such curve or line can be drawn by the continuous motion of a pencil in a sheet of paper. And Discontinuous function is just opposite of the continuous function , the function “f” is said to be discontinuous function when the graph drawn for f(x) is consists of disconnected curves or lines. For example: Continuous Function: Discontinuous Function: If we zoom into...
read moreBasic properties or theorems of limit.
The limit theorems or basic properties of limit are given below: 1> The limit of the sum (or difference) of the functions “f” and “g” is the sum (or difference) of the limits of the functions i.e. 2> The limit of the product of the function “f” and “g” is the product of the limits of the functions. i.e. 3> The limit of the quotient of the function “f” and “g” is the quotient of the limits of the functions. i.e. 4> The limit of nth root of a function...
read moreRight hand and Left hand limit of a function.
Let an Interval be denoted by (a-β , a+β) which is shown by the figure below: and x∈ (a-β , a+β) And let a function f(x) be defined at the Interval (a-β , a+β) . Then we can also find the limit of function f(x) as, Left hand Limit of a Function: In the above case the limit of f(x) when “x” approaches “a” from the left hand side of the interval is known as the left hand limit of f(x). and is denoted by: Right hand Limit of a Function: Similarly, the limit of f(x) when...
read moreInterval.
What is Interval? A set of points lying between any two points “a” and “b” is known as an Interval. For example: the interval from points x=1 to x=10 is the set of points lying between 1 and 10. Open and Closed Interval: An Interval which includes it’s end points(a and b) is known as Closed interval ; While an Interval Which does not includes it’s end points(a and b) is known as Open Interval. A open Interval from point “a” to point “b” is denoted by: (a,b) and A closed...
read moreThe concept of Limit.
What is limit? First study the following picture: You can see in the figure ; if a polygon is inscribed inside a circle , then as we increase the number of sides of polygon the area of polygon gradually increases . As we increase the number of sides in the polygon it’s area gradually approaches the area of circle and if we take a polygon with sufficiently large number of sides in same configuration then , the area of the polygon will become almost equal to the area of circle , but it will never be equal to the area of the...
read moreBasic properties of Logarithms.
Logarithms are important throughout mathematics and science. Establishing some basic properties or relations of Logarithms helps us for faster computation of mathematical problems. Some of the basic properties of Logarithms and proof of them are given below, they are also called theorems of logarithm. 1> For any positive x ,y and a: Proof: Suppose: then, and so, thus 2> For any positive a and x: Proof: suppose, then and thus, 3> For any positive x , y and a: proof: 4> For any positive a,b and x: Proof: suppose , then...
read moreThe Logarithmic Function.
The Logarithm. If “x” is a number and, Then , “y” is known as the logarithm of “x” to the base “a”. For example: :-The logarithm of 16 to the base 2 is 4 which can be shown as: :-The logarithm of 8 to the base 2 is 3 which can be shown as: The logarithmic Function: The function in which the relationbetween input(let x) and and output(let y) is given by: , where “a” is a constant, is known as the logarithmic function. when, , “y” is known as the logarithmic function...
read moreExponential Function.
Exponential Function: Exponential function is the function which is defined by the following formula: y=f(x)=ax Where , a is a constant great than 0 and a , x both are real numbers. In any exponential function defined by formula y=f(x)=ax, “a” is said to be the base of exponential function “f” and “x” is the exponent of “a”. Some Examples of Exponential Function: a> if f:A→B is defined by f(x)=2x then “f” is exponential function of base 2. We can show this exponential function...
read moreExponent(index) and laws of exponents(indices).
Exponent or Index: When a number (let a) is multiplied by itself multiple time(let n times) then we represent this case by adding “n” superscription to “a” as given below: an When , a number is repeatedly multiplied by itself the number of times the number is multiplied by itself is called the exponent the the number which is being multiplied is called base or index. For example: In an “a” the index or base and “n” is exponent of the base “a”. Laws of Exponents or Indices: The rules which gives...
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