Limit And Continuity In Calculus

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,

 And in Left Hand Limit the value of a, is negative for function f(x).The Mathematical Expression can be written as,

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.

∞x∞=∞

0x∞=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.