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

The limit of quotient of two functions as the input approaches some value is equal to quotient of their limits. It is called as quotient rule of limits and also called as division property of limits.

$x$ is a variable and two functions $f{(x)}$ and $g{(x)}$ are derived in terms of $x$. The limits of $f{(x)}$ and $g{(x)}$ as $x$ approaches to $a$ can be written 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, write limit of quotient of the functions $f{(x)}$ and $g{(x)}$ as $x$ tends to $a$ in mathematical form.

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

Evaluate the limit of division of the functions as $x$ tends to $a$ by replacing $x$ by $a$.

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

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

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

Therefore, it has proved that the limit of quotient of two functions as input approaches some value is equal to quotient of their limits. So, it is called as quotient rule of limits and also called as division property of limits.

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