A mathematical expression can be formed by the addition of two or more functions in mathematics. If you are a beginner, the sum of two or more functions definitely confuses you while finding its limit. So, a special limit formula is required for us in calculus to find limits of such functions.
There is a property of functions in mathematics and it can be applied here to find the limits of addition of two or more functions. Now, let’s know the addition property of functions in limits.
The limit of sum is equal to the sum of their limits. It is called the sum rule of limits
This addition rule of limits can be used to derive a formula in calculus and let’s express it in mathematical form.
Let’s denote two functions as $f{(x)}$ and $g{(x)}$. The functions can be added by addition and their sum is written as $f{(x)}+g{(x)}$ in mathematics.
Now, its limit as the value of variable $x$ approaches a value $a$ is written mathematically as follows.
$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big(f{(x)}+g{(x)}\Big)}$
According to sum law of limits in calculus, the limit of sum of functions $f(x)$ and $g(x)$ is equal to sum of the limits of $f(x)$ and $g(x)$ as $x$ tends to $a$.
$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big(f{(x)}+g{(x)}\Big)}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$ $+$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$
The above mathematical equation can be used as a formula in calculus. It is called in two ways as follows.
Now, let’s learn more about the limits’ addition rule from an understandable example. You have learned the addition rule of limits theoretically and it is time to know about it practically.
Evaluate $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize (x^2+3x)}$
Use the direct substitution method to find the limit of sum of functions.
$\implies$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize (x^2+3x)}$ $\,=\,$ $2^2+3(2)$
$\,\,=\,$ $2 \times 2+3 \times 2$
$\,\,=\,$ $4+6$
$\,\,=\,$ $10$
Now, let’s find the limit of each function in the expression.
$(1).\,\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize x^2}$ $\,=\,$ $2^2$ $\,=\,$ $4$
$(2).\,\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize 3x}$ $\,=\,$ $3(2)$ $\,=\,$ $6$
Add the limits of both functions to find their sum.
$\implies$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize x^2}$ $+$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize 3x}$ $\,=\,$ $4+6$ $\,=\,$ $10$
It is proved that the limit of sum of the functions is equal to the sum of their limits.
$\,\,\,\,\,\,\therefore\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize (x^2+3x)}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize x^2}$ $+$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize 3x}$ $\,=\,$ $4+6$ $\,=\,$ $10$
The limit problem in the above example verified the limit of a sum law mathematically.
The addition rule for limits is not limited to two functions and it can be extended to more than two functions. So, the limit of sum rule can be written generally as follows.
$\displaystyle \large \lim_{x \,\to\, a} \normalsize \Big(f_{1}{(x)}+f_{2}{(x)}+f_{3}{(x)}+\cdots\Big)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f_{1}{(x)}}$ $+$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f_{2}{(x)}}$ $+$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f_{3}{(x)}}$ $+$ $\cdots$
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