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Limits and Continuity.
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 even.
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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
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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 t
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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
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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(
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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 con
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