Q.Given $f(x)=\begin{cases}2x+1, & x<2\\ x^{2}-1, & x\ge 2\end{cases}$, find the left-hand limit and the right-hand limit of $f(x)$ at $x=2$, and state whether $\displaystyle\lim_{x\to2}f(x)$ exists.

$0.3$ neighbourhood of $3$ $=\{x:|x-3|<0.3\}$.

The $\delta$-neighbourhood of a point $a$ is the open interval $(a-\delta,\,a+\delta)$, i.e. the set of all $x$ whose distance from $a$ is less than $\delta$: $|x-a|<\delta$. Here $a=3$ and $\delta=0.3$, so the modulus form is $|x-3|<0.3$.

$|x-10|<\tfrac1{10}$ $=$ the $\tfrac1{10}$-neighbourhood of $10$ $=(9.9,10.1)$.

The modulus inequality $|x-a|<\delta$ is exactly the $\delta$-neighbourhood of the point $a$, namely the open interval $(a-\delta,\,a+\delta)$. Here $a=10$ and $\delta=\tfrac1{10}=0.1$, so it is the $\tfrac1{10}$-neighbourhood of $10$:

$\left(10-\tfrac1{10},\ 10+\tfrac1{10}\right)=(9.9,\ 10.1).$

$|x - 5| < 0.25$ is the $\delta$-neighbourhood of $5$ with radius $0.25$, written $N(5, 0.25)$ — option (d).

GSEB Class-12 Statistics, Limit chapter:

The neighbourhood notation is defined as:

$N(a, \delta) = \{x : |x - a| < \delta\}$

where $a$ is the centre and $\delta$ the radius. Comparing with:

$|x - 5| < 0.25$

we get $a = 5$ and $\delta = 0.25$, so the neighbourhood form is $N(5, 0.25)$.

The $\delta$-neighbourhood of $a$ is $|x-a|<\delta$; with $a=-5,\ \delta=0.001$ this is $|x+5|<0.001$.

The $\delta$-neighbourhood of a point $a$ consists of all $x$ satisfying $|x-a|<\delta$. Here $a=-5$ and $\delta=0.001$, so

$|x-(-5)|<0.001\;\Rightarrow\;|x+5|<0.001.$

The $\delta$ neighbourhood of $a$ is $\{x : |x-a|<\delta\} = (a-\delta,\ a+\delta)$ for some $\delta>0$.

Definition. For a real number $a$ and a positive number $\delta$, the $\delta$ neighbourhood of $a$ is the set of all real numbers $x$ whose distance from $a$ is less than $\delta$:

$\{x : |x-a|<\delta\} = (a-\delta,\ a+\delta).$

It is the open interval centred at $a$ with radius $\delta$.

"$x$ tends to $0$" — $x$ takes values as close to $0$ as we like, from both sides, but never equals $0$.

The symbol $x \to 0$ is read as "$x$ tends to $0$" or "$x$ approaches $0$." It means we let $x$ take a succession of values that come nearer and nearer to $0$:

• from the right ($x \to 0^{+}$): values like $0.1, 0.01, 0.001, \dots$
• from the left ($x \to 0^{-}$): values like $-0.1, -0.01, -0.001, \dots$

In this process $x$ can be made as close to $0$ as we please, but $x$ is never given the value $0$ itself. This idea of "approaching without reaching" is what makes a limit meaningful even where the function may be undefined at the point.