A number $L$ is said to be limit of a function $ƒ(x)$ at $x=a$ if $ƒ(x)$ approaches to $L$ whenever $x$ approaches to $a$. Let’s dive into more detailed explanation of the limit of a function starting from value of a function.
Value of a Function
If $ƒ$ is a function from $X$ to $Y$ and $x=a$ is an element in the domain of $ƒ$, then the image $ƒ(a)$ corresponding to $x=a$ is said to be the value of the function at $x=a$.
If the value of the function $ƒ(x)$ at $x=a$ denoted by $ƒ(a)$ is a finite number, then $ƒ(x)$ exists or is defined at $x=a$. If the value is not a finite number, then $ƒ(x)$ does not exist or is not defined at $x=a$.
If $y=ƒ(x)=2x+3$ at $x=2$ then, $y=ƒ(2)=2×2+3=7$. $7$ is a finite number. Hence, $ƒ(x)$ exists or is defined at $x=2$.
If $y=ƒ(x)=\frac{1}{x-1}$ at $x=1$, then, $y=ƒ(1)=\frac{1}{0}$. $\frac{1}{0}$ is not a finite number. Hence, $ƒ(x)$ does not exist or is not defined at $x=1$.
Meaning of $x\to a$
Let $x$ be a variable and consider $x\to 2$. Let us make the variable $x$ to take the values $1.9, 1.99, 1.999, 1.9999, …$
As the number of $9$’s increases, the value of $x$ will be nearer and nearer to $2$ but will never be $2$. In this situation, the numerical difference between $x$ and $2$ will be sufficiently small.
Again, let us make the variable $x$ to take the values $2.1, 2.01, 2.001, 2.0001, …$
As the number of zeros increases, the value of $x$ will be nearer and nearer to $2$ but will never be $2$. In this situation also, the numerical difference between $x$ and $2$ will be sufficiently small. Thus, if $x$ takes the value greater than $2$ or less than $2$ but the numerical difference between $x$ and $2$ is sufficiently small, then we say that $x$ approaches $2$ or $x$ tends to $2$ and we write it as $x\to 2$.
Idea of Limit
Lets take an example of a sequence of numbers $0.9, 0.99, 0.999, …$
In the sequence, the terms are gradually increasing but they always remain less than $1$. By making a proper choice of the term, we can make the term sufficiently close to $1$ or make the difference between $1$ and the term very small. In such a case, we say that the sequence tends to the limiting value $1$.
The above sequence can also be put in a functional notation by defining the function $ƒ$ by
\[\begin{array}{c} ƒ(1)=0.9 & \text{ 1st term} \\ ƒ(2)=0.99 & \text{ 2nd term} \\ ƒ(3)=0.999 & \text{ 3rd term} \\ …………… \\ ƒ(n)=0.99…9 \text{ (n 9’s)} & \text{ nth term} \end{array}\]
Hence, when $n$ tends to infinity, $ƒ(n)$ becomes almost equal to $1$. So, the limiting value of $ƒ(n)$ is $1$ and it is denoted by \[\lim_{n \to \infty} ƒ(n)=1\]
Limit of a Function
Consider a function $y=ƒ(x)=2x+1$. If we consider the sequence of values of $x$ to be $0.5, 0.75, 0.9, 0.99, 0.999, …$ whose limit is $1$, we find the corresponding values of $ƒ(x)$ to be $2, 2.5, 2.8, 2.98, 2.998, …$ which go nearer and nearer to $3$ when $x$ goes nearer to $1$. So, when $x$ is sufficiently close to $1$, $ƒ(x)$ is close to $3$.
Again, if we consider the sequence of values of $x$ to be $1.5, 1.25, 1.1, 1.01, 1.001, 1.0001, …$ whose limit is $1$, we find the corresponding values of $ƒ(x)$ to be $4, 3.5, 3.2, 3.02, 3.002, 3.0002, …$ which go nearer and nearer to $3$ when $x$ goes nearer to $1$.
Thus, when $x\to 1$, $ƒ(x)\to 3$ and we write \[\lim_{x \to 1} ƒ(x)=\lim_{x \to 1} (2x+1)=3\] Hence, a number $L$ is said to be limit of a function $ƒ(x)$ at $x=a$ if $ƒ(x)$ approaches to $L$ whenever $x$ approaches to $a$ and we write \[\lim_{x \to a } ƒ(x)=L\]
A neighborhood of a point $a$ is an open interval containing the point $a$. It is generally denoted by $(a-δ, a+δ)$. \[x∈(a-\delta, a+\delta) \Rightarrow |x-a|<\delta\] \[\lim_{x \to a} ƒ(x)=L\]
If to every positive number $ε$, however small, there corresponds a positive number $δ$, such that \[|ƒ(x)-L|<ε, \text{ whenever } |x-a|<δ\]
Left Hand Limit (LHL) and Right Hand Limit (RHL)
A number $L$ is said to be left hand limit of a function $ƒ(x)$ at $x=a$ if $ƒ(x)$ approaches to $L$ whenever $x$ approaches to $a$ through value less than $a$ (i.e. $x$ approaches $a$ from left) and we write \[\lim_{x \to a^-} ƒ(x)=L\]
A number $L$ is said to be right hand limit of a function $ƒ(x)$ at $x=a$ if $ƒ(x)$ approaches to $L$ whenever $x$ approaches to $a$ through value greater than $a$ (i.e. $x$ approaches $a$ from right) and we write \[\lim_{x \to a^+} ƒ(x)=L\]
The limit of a function $ƒ(x)$ at $x=a$ exists only when both LHL and RHL exist and coincide. That is, $\lim_{x \to a} ƒ(x)$ exists if and only if \[\lim_{x \to a^-} ƒ(x)=\lim_{x \to a^+} ƒ(x)\]
Meaning of Infinity
Consider a function \[y=ƒ(x)=\frac{1}{x}\] Considering the sequence of values of $x$ to be $1, 0.5, 0.1, 0.01, 0.001, 0.0001, …$ whose limit is $0$, we find the corresponding values of $ƒ(x)$ to be $1, 2, 10, 100, 1000, 10000, …$ which go on increasing. If the value of $x$ is very small, then the corresponding value of $ƒ(x)$ will be very large. If we take the value of $x$ to be sufficiently close to $0$, the value of $ƒ(x)$ will be greater than any positive number, however large. In such case, $ƒ(x)\to\infty$ as $x\to 0$ and we write \[\lim_{x \to 0}\frac{1}{x}=\infty\]
Hence, if the value of $ƒ(x)$ is greater than any pre-assigned number, however large, by making $x$ sufficiently close to $a$, we say that the limit of $ƒ(x)$ is infinity as $x$ tends to $a$. \[\lim_{x \to a} ƒ(x)=\infty\]
If $x$ tends to $∞$
In the same function, \[y=ƒ(x)=\frac{1}{x}\]
We see that when the value of $x$ increases, the corresponding value of $ƒ(x)$ decreases. If we take the sequence of values of $x$ to be $1, 10, 100, 1000, 10000, …$ we find that the corresponding values of $ƒ(x)$ to be $1, 0.1, 0.01, 0.001, 0.0001, …$ which go on decreasing. Hence, if we take the value of $x$ to be sufficiently large (the value greater than any positive number, however large) the value of $ƒ(x)$ will be sufficiently close to $0$ and we write \[\lim_{x \to \infty} ƒ(x)=\lim_{x \to \infty} \frac{1}{x}=0\]
Hence, if $ƒ(x)$ can be made sufficiently close to $L$ when $x$ is greater than any pre-assigned number, however large, then we say that the limit of the function $ƒ(x)$ is $L$ when $x\to \infty$. \[\lim_{x \to \infty} ƒ(x)=L\]
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