Let $f(x)$ and $g(x)$ be two functions in terms of $x$. The composition of them forms a composite function $f{\Big(g{(x)}\Big)}$. The limit of the composite function as $x$ approaches $a$ is written as follows in mathematics.
$\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{\Big(g{(x)}\Big)}}$
Let us learn how to derive the limit rule to find the limit of the composite function.
The limit of the function $g(x)$ as the value of $x$ closer to $a$ is written in mathematical form as follows.
$\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$
Let’s use the direct substation method to find the limit of the function $g(x)$ as $x$ tends to $a$.
$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$ $\,=\,$ $g(a)$
$\,\,\,\therefore\,\,\,\,\,\,$ $g(a)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$
The limit of the composite function $f{\Big(g{(x)}\Big)}$ as $x$ approaches $a$ is expressed in the following form in calculus.
$\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{\Big(g{(x)}\Big)}}$
Use the direct substation method and find evaluate the limit of the composite function as $x$ closer to $a$.
$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{\Big(g{(x)}\Big)}}$ $\,=\,$ $f{\Big(g{(a)}\Big)}$
In the previous step, we have evaluated the following mathematical equation.
$\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{\Big(g{(x)}\Big)}}$ $\,=\,$ $f{\Big(g{(a)}\Big)}$
In the first step, the limit of the function $g(x)$ is evaluated when the value of $x$ approaches $a$.
$g(a)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$
Now, substitute the value of g(a) in the above mathematical equation to prove a limit rule for finding the limit of a composite function.
$\,\,\,\therefore\,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{\Big(g{(x)}\Big)}}$ $\,=\,$ $f{\Big(\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}\Big)$
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