# 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 $\Big(f(x)\Big)^n$. 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.

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

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 $\Big(f(x)\Big)^n$ as $x$ tends to $a$ in mathematical form.

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

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

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

### Power Rule of Limits in mathematical form

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

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

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

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

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.

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