Power Rule of Limits

Formula

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

What is Limits’ Power rule?

The functions in general form appear with exponents in some special cases and they are called power functions. A special limit formula is required to find the limit of power of a function and it is called the power rule of limits.

A function is written in general form as $f(x)$ mathematically, where the literal $x$ denotes a variable and its power function can be written as $\big(f(x)\big)^{\displaystyle n}$ where the literal $n$ denotes a constant. Now, the limit of a power function as the variable $x$ approaches to a value $a$ is written in calculus as follows.

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

According to the limits power law, the limit of power of a function is equal to the power of its limit. So, the limit of a function with $n$ as its power as the variable $x$ approaches to a value $a$ is equal to the limit of function as $x$ tends to $a$ raised to the power of $n$.

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

The above explanation on power rule in limits helps you to know what the limits law on power functions. Now, let’s see some examples to understand how to use the limits’ power property in calculus.

Examples

$(1).\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize {(x+6)}^2}$ $\,=\,$ $\Big(\displaystyle \large \lim_{x \,\to\, 0}{\normalsize {(x+6)}\Big)}^2$

$(2).\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize {(x^2+3x+4)}^{-5}}$ $\,=\,$ $\Big(\displaystyle \large \lim_{x \,\to\, 2}{\normalsize {(x^2+3x+4)}\Big)}^{-5}$

$(3).\,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize {\bigg(7+\dfrac{2}{x}\bigg)}^{\Large \frac{3}{2}}}$ $\,=\,$ $\Bigg(\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize {\bigg(7+\dfrac{2}{x}\bigg)}\Bigg)^{\Large \frac{3}{2}}}$

You have clearly learned the power law of limits and it is time to learn the derivation of the limits power rule’s formula.