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

The limit of product of two or more functions as the input approaches some value is equal to product of their limits. It is called as product rule of limits and also called as multiplication property of limits.

$x$ is a variable and two functions $f{(x)}$ and $g{(x)}$ are defined in terms of $x$. The limits of $f{(x)}$ and $g{(x)}$ as $x$ tends to $a$ are expressed mathematically as follows.

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$ $\,=\,$ $f{(a)}$

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$ $\,=\,$ $g{(a)}$

Now, express limit of product of the function $f{(x)}$ and $g{(x)}$ as $x$ approaches $a$ in mathematical form.

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

Find the limit of product of functions as $x$ approaches $a$ by substituting $x$ is equal to $a$.

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

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

Finally, replace the limits $f{(a)}$ and $g{(a)}$ in limit form.

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

Therefore, it is proved that the limit of product of two functions as input approaches some value is equal to product of their limits. It is called as product rule of limits and can also be called as multiplication property of limits.

Remember, the law of product rule of limits is not limited to two functions and it can used more than two functions.

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

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