The difference of limits rule is an important limit operation in calculus and it expresses that the limit of difference between two functions is equal to the difference between their limits. Now, let’s learn how to prove the limits subtraction law in mathematics.
Let’s consider two functions, whereas each function is defined in terms of $x$ and the functions are written as $f(x)$ and $g(x)$ mathematically. Let’s assume that the function $g(x)$ is subtracted from $f(x)$, and the difference between them is written as $f(x)-g(x)$ in mathematics.
The limit of subtraction of functions as its variable $x$ approaches a value $a$, is written as follows.
$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \big[f(x)-g(x)\big]}$
Now, let’s know the process of proving the limits difference rule by finding the limit of the difference.
The limits by direct substitution method can be used to find the limit. So, substitute $x$ equals to $a$ in the function to find the limit of subtraction of functions as the value of $x$ approaches to $a$.
$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \big[f(x)-g(x)\big]}$ $\,=\,$ $f(a)-g(a)$
Now, let’s find the limit of both functions by direct substitution method as the value of $x$ tends to $a$.
Substitute $x = a$ in function $f(x)$ to find the limit of function $f(x)$ as $x$ approaches to $a$.
$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$ $\,=\,$ $f(a)$
The limit of a function $f(x)$ as the value of $x$ tends to $a$ is $f(a)$. So, it can be written as follows.
$\,\,\,\,\,\,\therefore\,\,\,$ $f(a)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$
Similarly, substitute $x = a$ in the function $g(x)$ to find the limit of function $g(x)$ as the value of $x$ closer to the value of $a$.
$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}$ $\,=\,$ $g(a)$
The limit of a function $g(x)$ as the value of $x$ tends to $a$ is $g(a)$. So, it can also be written as follows.
$\,\,\,\,\,\,\therefore\,\,\,$ $g(a)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}$
Now, let’s find the relationship between the functions for proving the subtraction law for limits in calculus.
We have evaluated above that
$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \big[f(x)-g(x)\big]}$ $\,=\,$ $f(a)-g(a)$
We have also evaluated above that
$(1).\,\,$ $f(a)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$
$(2).\,\,$ $g(a)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}$
Now, substitute the values of $f(a)$ and $g(a)$ in the above equation.
$\,\,\,\,\,\,\therefore\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \big[f(x)-g(x)\big]}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$ $-$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}$
Therefore, it is proved that the limit of difference between two functions is equal to the difference between their limits. It is called the difference rule of limits and it is also called the subtraction rule of limits.
The multiplication rule of limits with proof to learn how to find the limit of a product of functions.
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