Math Doubts

Addition Rule of Limits

Formula

$\displaystyle \large \lim_{x \,\to\, a} \normalsize \Big[f{(x)}+g{(x)}\Big]$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$ $+$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$

The limit of sum of functions is equal to the sum of their limits. It is called as sum rule of limits and also called as the addition rule of limits.

Proof

$f(x)$ and $g(x)$ are two different functions in terms of $x$. The sum of the functions is $f(x)+g(x)$.

Find limit of sum of the functions

Take the value of $x$ approaches $a$. The limit of sum of the functions as $x$ approaches $a$ is written in calculus in the following mathematical form.

$\displaystyle \large \lim_{x \,\to\, a} \normalsize \Big[f(x)+g(x)\Big]$

Now, substitute $x = a$ to find the value of sum of the functions when $x$ tends to $a$.

$\implies$ $\displaystyle \large \lim_{x \,\to\, a} \normalsize \Big[f(x)+g(x)\Big]$ $=$ $f(a)+g(a)$

Find limit of each function

Now, evaluate limit of each function as $x$ approaches $a$

$\displaystyle \large \lim_{x \,\to\, a} \normalsize f(x) = f(a)$

$\displaystyle \large \lim_{x \,\to\, a} \normalsize g(x) = g(a)$

Compare results of both steps

According to the result of the first step,

$\implies$ $\displaystyle \large \lim_{x \,\to\, a} \normalsize \Big[f(x)+g(x)\Big]$ $=$ $f(a)+g(a)$

But $f(a) = \displaystyle \large \lim_{x \,\to\, a} \normalsize f(x)$ and $g(a) = \displaystyle \large \lim_{x \,\to\, a} \normalsize g(x)$ as per result of second step.

$\therefore \,\,\,\,\,\, \displaystyle \large \lim_{x \,\to\, a} \normalsize \Big[f(x)+g(x)\Big]$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a} \normalsize {f(x)} + \displaystyle \large \lim_{x \,\to\, a} \normalsize {g(x)}$

It is proved that the limit of sum of the functions is equal to the addition of their limits.

The sum law of limits can also be extended to more two functions as well. If $f_{1}(x)$, $f_{2}(x)$, $f_{3}(x) \ldots$ are functions, 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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