Math Doubts

Limit

The value of a function as its input approaches some value is called the limit.

Introduction

According to English language, one of the meanings of “Limit” is a value. In calculus, the limit represents a value of a function as the input of the function approaches to some value.

$f{(x)} = 2x+3$ is a function.
Find the value of this function at a point. For example, $x = 5$.

$\implies$ $f{(5)} = 2(5)+3 = 13$

Therefore, the value of the function is $13$ at $x$ equals to $5$.

Let’s evaluate the limit of the function as $x$ approaches $5$. Firstly, it is essential to know the difference between $x$ approaches $5$ and $x$ equals to $5$ for understanding the concept of limit.

  1. The meaning of $x$ approaches $5$ is, it’s actually a number closer to $5$ but either less than or slightly greater than $5$. In other words, $x \approx 5$ but $x < 5$ or $x > 5$. For example, $x = 4.99, 4.992, 4.9919$ and so on ($x < 5$) and $x = 5.00001, 5.0005, 5.002,$ and so on. ($x > 5)$ but the approximate value of each number is $5$.
  2. The meaning of $x$ equals to $5$ is, it’s a number exactly equals to $5$ ($x = 5$). It’s neither less than $5$ nor greater than $5$.

Now, calculate the values of the above function for the $x$ values closer to $5$ and then compare each value with the value of the function for the $x = 5$.

$x$ $f{(x)} = 2x+3$ $Approximate$
$4.9$ $12.8$ $12.8 \ne 13$
$4.99$ $12.98$ $\approx 13$
$4.999$ $12.998$ $\approx 13$
$4.9999$ $12.9998$ $\approx 13$
$4.99999$ $12.99998$ $\approx 13$
$\vdots$ $\vdots$ $\vdots$
$5$ $13$ $13$
$\vdots$ $\vdots$ $\vdots$
$5.00001$ $13.00002$ $\approx 13$
$5.0001$ $13.0002$ $\approx 13$
$5.001$ $13.002$ $\approx 13$
$5.01$ $13.02$ $\approx 13$
$5.1$ $13.2$ $13.2 \ne 13$

The values of the function are $12.8$ and $13.2$ for $x = 4.9$ and $x = 5.1$ respectively but they both are not equal to the value of the function for $x = 5$ because $x = 4.9$ and $5.1$ are closer to $x = 5$ but not very closer.

As the value of $x$ increases closer to $5$ but not equal to $5$ then each value of $f{(x)}$ is also closer to the value of the function for $x = 5$. Similarly, if you take $x$ value negligibly greater than $5$, then also each value of function is closer to the value of the function for $x = 5$.

Any one of the values of the function for $x$ closer to $5$ is called the limit of function as $x$ approaches $5$ but don’t think that the limit is the value of the function for $x$ equals to $5$ because there is a lot of difference between $x$ equals to $5$ and $x$ approaches $5$.

Due to negligible difference between the values of the function for $x$ approaches $5$ and $x$ equals to $5$, the limit of the function as $x$ approaches $5$ can be taken as the value of the function for $x$ equals to $5$. That’s why, we calculate the limit of any function by considering its input approaching value as the input value of the function in calculus.

Representation

Mathematically, the operation of finding limit is symbolically represented by $\lim$ and the input approaches to certain value is displayed as subscript to $\lim$ symbol. Therefore, the limit of a function as its input approaches to some value is mathematically expressed as follows.

$\displaystyle \large \lim_{x \,\to\, 5}{\normalsize (2x+3)}$ $\,=\,$ $13$

This mathematical equation is read in two ways in calculus. You can follow any one of them.

  1. The limit of $2x+3$ as $x$ approaches $5$ is equal to $13$
  2. The limit of $2x+3$ as $x$ tends to $5$ is equal to $13$


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