Proof of Composite Limit rule

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.

Calculate the Limit of the 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 $a$.

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

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

Find the Limit of composite function

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 $a$.

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

Identify the Relation between Limits

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.

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