Non - Removable, Infinite and Oscillatory Type Discontinuity - Practice Questions & MCQ

Non - Removable, Infinite and Oscillatory Type Discontinuity

Non - Removable Discontinuity

In this type of discontinuity, the limit of the function at $\mathbf{x}=$ a does not exists, i.e. $\lim _{x \rightarrow a} f(x)$ does not exists, so it is not possible to redefine the function in any way to make it continuous at $\mathrm{x}=\mathrm{a}$.

Again non-removable discontinuity can be classified into three categories
1. Finite Discontinuity

In this type of discontinuity, both left hand and right hand limit exists but, they are unequal.
i.e. $\lim _{x \rightarrow a^{-}} f(x)=L_1 \quad$ and $\quad \lim _{x \rightarrow a^{+}}=L_2$ but $L_1 \neq L_2$

For example

$
f(x)=\left\{\begin{array}{cc}
x^2, & x \leq 1 \\
x+1, & x>1
\end{array}\right.
$
Here, $\quad \lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1^{+}}(x+1)=2$

$
\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{-}} x^2=1
$
At $x=1$, both LHL and RHL exist, but they are not equal

The value of function jumps by " 1 " units at $x=1$, hence, such kinds of discontinuity are also known as jump discontinuity.
2. Infinite Discontinuity

Here, at least one of the two limits (L.H.L. and R.H.L) is infinity or minus infinity
Consider the function,

$
f(x)=\frac{1}{x}
$
Here, L.H.L. $=\lim _{x \rightarrow 0^{-}} \frac{1}{x}=-\infty$ and R.H.L. $=\lim _{x \rightarrow 0^{+}} \frac{1}{x}=\infty$

Function has infinite type of discontinuity at $x=0$
3. Oscillatory Type Discontinuity

In this type of discontinuity, the limit of the function doesn't exists but oscillates between two quantities.

For example,

$
f(x)=\sin \left(\frac{1}{x}\right)
$

$\lim _{x \rightarrow 0} f(x)=$ a value between -1 to 1
$\therefore$ limit doesn't exists as it's oscillates between -1 to 1 as $x \rightarrow 0$

Graph of $f(x)=\sin (1 / x)$