In mathematics, a mathematical expression can be formed by the subtraction of two functions. If you are a beginner, the difference between the functions definitely confuses you while you try to find its limit. So, a special limit formula is required for us to find the limits of such functions in calculus.
A property of functions is there in mathematics and it can be used here to find the limits of difference between any two functions. Now, let’s know the subtraction property of functions in limits.
The limit of difference is equal to difference between their limits. It is called difference rule of limits
The limits’ subtraction property can be used to derive a limit formula in calculus and let’s express it in mathematical form.
Let’s write two functions as $f{(x)}$ and $g{(x)}$. If a function $g{(x)}$ is subtracted from $f{(x)}$ and then their difference is written as $f{(x)}-g{(x)}$ mathematically.
Now, its limit as the value of variable $x$ approaches to a value $a$ is written in mathematics as follows.
$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big(f{(x)}-g{(x)}\Big)}$
According to the subtraction law of limits in calculus, the limit of difference between the functions $f(x)$ and $g(x)$ is equal to the difference between the limits of $f(x)$ and $g(x)$ as $x$ tends to $a$.
$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big(f{(x)}-g{(x)}\Big)}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$ $-$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$
This mathematical equation can be used as a formula in calculus. It is called in two ways as follows.
Now, It is time to know more about the limits’ subtraction law from an understandable example here. You learned the subtraction rule of limits theoretically and let’s know about it practically.
Evaluate $\displaystyle \large \lim_{x \,\to\, 4}{\normalsize (x^2-2x)}$
Use the direct substitution method to find the limit of subtraction of functions.
$\implies$ $\displaystyle \large \lim_{x \,\to\, 4}{\normalsize (x^2-2x)}$ $\,=\,$ $4^2-2(4)$
$\,\,=\,$ $4 \times 4-2 \times 4$
$\,\,=\,$ $16-8$
$\,\,=\,$ $8$
Now, let’s evaluate the limit of every function of the expression.
$(1).\,\,$ $\displaystyle \large \lim_{x \,\to\, 4}{\normalsize x^2}$ $\,=\,$ $4^2$ $\,=\,$ $16$
$(2).\,\,$ $\displaystyle \large \lim_{x \,\to\, 4}{\normalsize 2x}$ $\,=\,$ $2(4)$ $\,=\,$ $8$
Finally, subtract the limits of both functions to find their difference.
$\implies$ $\displaystyle \large \lim_{x \,\to\, 4}{\normalsize x^2}$ $-$ $\displaystyle \large \lim_{x \,\to\, 4}{\normalsize 2x}$ $\,=\,$ $16-8$ $\,=\,$ $8$
It is proved that the limit of difference between the functions is equal to the difference between their limits.
$\,\,\,\,\,\,\therefore\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 4}{\normalsize (x^2-2x)}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, 4}{\normalsize x^2}$ $-$ $\displaystyle \large \lim_{x \,\to\, 4}{\normalsize 2x}$ $\,=\,$ $16-8$ $\,=\,$ $8$
The limit problem in the above example proved the limit of a difference rule.
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