The limit of a function is an interesting but a little bit complex concept where, it may be possible that we try to find the value in the neighborhood of the Point of a function because the value of the function does not exist at a point. On the other hand, continuity of a function is closely related to the concept of limits. Now, talk about the definition of Limit, in many cases the values of the given function ‘f’ for the values of ‘x’ near a point ‘c’ and this value is not equal to function f (c) value or that value lies near no number at all.
Now, next concept to study is how we represent the limit, let a function 'f' is said to tends to a and the limit as 'x' tends to 'a', if x=a then it shows the values is just larger than or just smaller than x=a, f(x) has to move more closely to the value of limit and mathematically it can be written as,
Lim x→ a f(x) = 1, which is equal to
|f(x)-l| < e, x: 0 < |x-a| <d
Where, e and d are positive Numbers.
There are two types of limit that is Right and Left hand limits, in the Right Hand Limit
Lim x→ a+f(x) = 1, the value of a, is positive for function f(x).For example if the value of x tends to 1 than,
Lim x→ 1 (x) = 1,
Lim x→ a-f(x) = 1,
For example if x tends to the value -1 than
Lim x→ 1-(x) = 0,
Now talk about infinity, infinity means something which keeps increasing and passes all limits this is called positive infinity. On the other side, something that continuously decrease and passes all limits is called negative infinity. The symbol of infinity is ∞. Some points given below which are necessary to read.
Infinity cannot be plotted on the paper.
∞-∞ is indeterminate.
∞/∞,∞0 , are all indeterminate.
Now the definition of continuity, the any function that is f(x) is called continuous on an interval a if
Lim x→ a f(x) = f(a),
Otherwise, the function f(x) is discontinuous at a. Note that the continuity of f(x) at a means two things,
Lim x→ a f(x) the function exist
And this limit is f(a).
Lim x→ a f(x) = f(a)
This property is known as continuity for all Real Numbers a. The f(x) is said to be continuous from the left if the value of a is negative that is
Lim x→a- f(x) = f(a),
And the f(x) is said to be continuous from the right if the value of a is positive that is
Lim x→a+f(x) = f(a)
This shows the continuity and limits.
There are some points of continuity given below, if f(x) and g(x) are continuous at a, Then
(1) f(x) + g(x) is continuous.
(2) f(x) g(x) is continuous at a
(3) f(x)/g(x) is continuous at a g(a)≠0.
A function is said to be continuous if function f ( x ) at Point ( c , f ( c ) ) if the conditions listed below are satisfied-
1) Function f ( c ) must exist.
2) The lim x → c f ( x ) must also exist.
3) And lim x → c f ( x ) = f ( c ).
The meaning of above conditions function is that there should be no missing point or gaps for f (x...Read More
The continuity of a Functions can be determined by the fact that their graphs are continuous in nature i.e. the graph of the function is continuous. Continuous graphs are the type of the graphs in which there is no break in the graph and which are drawn without lifting the pencil from the paper. In the term of continuous Functions, it is also tr...Read More
A function f(x) is said to be continuous at x= a if limx→a- f(x) = limx→a+ f(x)=f(a).
i.e. if Left Hand Limit = Right hand limit = value of the function at 'a'.
i.e. limx→a f(x) = f(a),
If, f(x) is not continuous at x = a, we say that f(x) is discontinuous at x = a.
For a function to be continuous at any Point x=a,...Read More
When we deal with Calculus we have to deal with Continuous Calculus and Discontinuous Calculus. In Continuous Calculus, we focus on continuous Functions and in discontinuous Calculus we deal with discontinuous Functions. Discontinuous functions are those functions which are not continuous means a function whose values does not vary continuously w...Read More
Continuous Functions are those Functions which have no breaks i.e. when we draw them on a graph it seems to be a smooth curve or graph from one end to the other end without any cut or break between it. To understand Discontinuity we need to learn different types of discontinuous functions.