$\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{\Big(g{(x)}\Big)}}$ $\,=\,$ $f{\Big(\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}\Big)$
Suppose $f(x)$ and $g(x)$ represent two functions in terms of $x$. The composition of them is expressed as $f{\Big(g{(x)}\Big)}$ in mathematics. In calculus, the limit of a composite function as $x$ approaches $a$ is written as follows.
$\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{\Big(g{(x)}\Big)}}$
The limit of $f$ of $g$ of $x$ as $x$ closer to $a$ is equal to $f$ of the limit of $g$ of $x$ as $x$ tends to $a$.
$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{\Big(g{(x)}\Big)}}$ $\,=\,$ $f{\Big(\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}\Big)$
This mathematical equation is called the composite limit rule and it is used to find the limit of a function, which is formed by the composition of two or more functions.
Let’s verify the composite limit rule from the following understandable example problem.
Evaluate $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \cos{\big(x^2-4\big)}}$
Now, find the limit of the cosine of angle $x$ square minus $4$ as $x$ approaches $2$ by the direct substation.
$=\,\,\,$ $\cos{\big((2)^2-4\big)}$
$=\,\,\,$ $\cos{(4-4)}$
$=\,\,\,$ $\cos{(0)}$
$=\,\,\,$ $1$
Similarly, find the cosine of the limit of the function $x$ squared minus $4$ as $x$ closer to $2$ by the direct substitution method.
$=\,\,\,$ $\cos{\Big(\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \big(x^2-4\big)\Big)}}$
$=\,\,\,$ $\cos{\Big(\big((2)^2-4\big)\Big)}$
$=\,\,\,$ $\cos{(4-4)}$
$=\,\,\,$ $\cos{(0)}$
$=\,\,\,$ $1$
$\,\,\,\therefore\,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \cos{\big(x^2-4\big)}}$ $\,=\,$ $\cos{\Big(\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \big(x^2-4\big)\Big)}}$ $\,=\,$ $1$
Learn how to derive the composition limit rule to find the limit rule of a composite function.
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