Proof of Addition of Limits Rule
Fact-checked:
The sum of limits rule is an important limit formula and it expresses that the limit of the sum of two or more functions is equal to sum of their limits. Let’s learn how to prove the limits addition law mathematically in calculus.
Let’s consider two functions in terms of $x$ and they are simply written as $f(x)$ and $g(x)$ in mathematics. The sum of the functions is written as $f(x)+g(x)$.
Addition
The limit of sum 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 start the process of proving the limits sum rule by finding the limit of a sum.
Step: 1
The limits by direct substitution method is a fundamental method in calculus. So, substitute $x$ equals to $a$ in the function to find the limit of the sum 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 each function by the direct substitution method as the value of $x$ tends to $a$.
Step: 2
Substitute $x = a$ in function $f(x)$ to find the limit of function $f(x)$ as the value of $x$ is closer 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 $x$ approaches $a$ 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 $x$ approaches 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 $x$ approaches $a$ is $g(a)$. So, it can also be written as follows.
$\,\,\,\,\,\,\therefore\,\,\,$ $g(a)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}$
It is time to find the relationship between the functions for proving the sum law for limits in calculus.
Step: 3
It is evaluated 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.
$\,\,\,\,\,\,\therefore\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \big[f(x)+g(x)\big]}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$ $+$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}$
It is proved that the limit of sum of two functions is equal to the sum of their limits. It is called the sum rule of limits and it is also called the addition rule of limits in calculus.
Extension
The limit sum law can be extended to more than two functions. So, the addition rule for limits can be written as follows and it can be proved by following the above procedure.
If the functions are written as $f_{1}{(x)}$, $f_{2}{(x)}$, $f_{3}{(x)} \ldots$ then the sum rule of limits is written as follows.
$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \big[f_{1}{(x)}+f_{2}{(x)}+f_{3}{(x)}\ldots\big]}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f_{1}{(x)}}$ $+$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f_{2}{(x)}}$ $+$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f_{3}{(x)}}$ $+$ $\ldots$
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