Proof of Power Rule of Limits

$x$ is a variable, $a$ and $n$ are constants. The function in terms of $x$ is written as $f(x)$. It formed a power function with a constant $n$ and the power function is written in mathematical form as $\Big(f(x)\Big)^{\displaystyle n}$. The limit of the $n$-th power of a function $f(x)$ as $x$ approaches $a$ is written in calculus in the following mathematical form.

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big(f(x)\Big)^{\displaystyle n}}$

The limit of the power function can be evaluated by the power rule of limits and it can be derived mathematically in calculus in three simple steps.

Evaluate the Limit of a function

Write, the limit of a function $f(x)$ as $x$ approaches $a$ in mathematical form.

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

Take, the limit of a function $f(x)$ as $x$ approaches $a$ is equal to $L$.

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

Now, find the limit of the function by the direct substitution method.

$\implies$ $L \,=\, f(a)$

Evaluate the Limit of a power function

Express the limit of a power function $\Big(f(x)\Big)^{\displaystyle n}$ as $x$ tends to $a$ in mathematical form.

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big(f(x)\Big)^{\displaystyle n}}$

Now, evaluate the limit of the power function as $x$ approaches $a$ by direct substitution method.

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

Power Rule of Limits in mathematical form

In the first step, the value of $f(a)$ is $L$.

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

But, it is taken that $L$ is equal to the $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$

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

Therefore, it is proved that the limit of a power function is equal to the power of the limit of the function. This property is called the power rule of limits.