Math Doubts

Proof of Power Rule of Limits

The limits power rule states that the limit of power of a function is equal to the power of its limit. Now, let’s learn how to derive the power law in limits mathematically.

$x$ 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.

Step1

Evaluate the Limit of a function

The limit of the function $f(x)$ as the value of $x$ approaches to $a$ is written in calculus as follows.

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

Now, let’s find the limit by using direct substitution method. So, substitute $x$ is equal to $a$ to find the limit.

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

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

Step2

Evaluate the Limit of Power function

The limit of function $f(x)$ raised to the power of $n$ as the value of $x$ tends to $a$ is written mathematically as follows.

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

Now, let’s find the limit of power function as $x$ approaches to $a$ by direct substitution method.

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

Step3

Revealing the Relations between the equations

We have derived two mathematical equations as per the above two steps.

$(1).\,\,\,$ $f(a)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$

$(2).\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \big(f(x)\big)^{\displaystyle n}}$ $\,=\,$ $\big(f(a)\big)^{\displaystyle n}$

Now, substitute the value of $f$ of $a$ from the first equation in the right-hand side expression of second equation.

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

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