Limit and Continuity
Class 10: Optional Math
Concept of Continuity, Continuity in Set of Numbers, Discontinuity in Graph, Notational Representation of Continuity
Notational Representation of Continuity
Left hand Limit
When x approaches to a point a from the left, then it is denoted by \(x \rightarrow a^-\) and the left hand limit of the function \(f(x)\) at a point \(a\) is defined by,
\(\displaystyle \lim_{x \to a^-} f(x) = L\)
means that as \(x\) gets closer and closer to \(a\) with \(x < a\), the values of \(f(x)\) approach \(L\)
Notation:
* \(\displaystyle \lim_{x \to a^-} f(x)\) → Limit of \(f(x)\) as \(x\) approaches \(a\) from the left.
* The “\(-\) ” sign on \(a^-\) indicates approaching from the left.
Right Hand Limit
When x approaches to a point a from the right, then it is denoted by \(x \rightarrow a^+\) and the left hand limit of the function \(f(x)\) at a point \(a\) is defined by,
\(\displaystyle \lim_{x \to a^+} f(x) = R\)
means that as \(x\) gets closer and closer to \(a\) with \(x > a\), the values of \(f(x)\) approach \(R\)
Notation:
* \(\displaystyle \lim_{x \to a^+} f(x)\) → Limit of \(f(x)\) as \(x\) approaches \(a\) from the right.
* The “\(+\) ” sign on \(a^+\) indicates approaching from the right.
Functional Value
If \(f\) is a function, then for each number x in its domain, the corresponding image in the range is designated by the symbol \(f(x)\), read as \(f\) of \(x\) or as \(f\) at \(x\). We refer to \(f(x)\) as the value of the function \(f\) at the number \(x\).
\(\displaystyle \lim_{x \to a} f(x) = f(a)\)
Means that the value at the exact point \(a\).
Condition for Continuity
For the function to be continuous at \(x = a\) the following condition must be satisfied.
- \(f(a)\) must exist.
- Left hand limit = Right Hand Limit
\(\displaystyle \lim_{x \to a^-} f(x) = \displaystyle \lim_{x \to a^+} f(x)\)
- The value of a function and the limit value must be equal to the left-hand limit and right-hand limit.
In conclusion,
