# Limit of Functions Involving nth Power

In this tutorial we shall discuss an example of evaluating limits involving a function with nth power of variable. In most cases if limit involves an nth power variable expression we solve using the binomial theorem.

Let us consider an example which
$\mathop {\lim }\limits_{x \to a} \frac{{{x^n} – {a^n}}}{{x – a}}$

Let us consider $x = a + h$, as $x \to a$, then $h \to 0$, we have

Now
$\begin{gathered} \mathop {\lim }\limits_{x \to a} \frac{{{x^n} – {a^n}}}{{x – a}} = \mathop {\lim }\limits_{h \to 0} \frac{{{{\left( {a + h} \right)}^n} – {a^n}}}{{a + h – a}} \\ \Rightarrow \mathop {\lim }\limits_{x \to a} \frac{{{x^n} – {a^n}}}{{x – a}} = \mathop {\lim }\limits_{h \to 0} \frac{{{{\left( {a + h} \right)}^n} – {a^n}}}{h} \\ \end{gathered}$

Using the binomial expression, we have
$\begin{gathered} = \mathop {\lim }\limits_{h \to 0} \frac{{{a^n} + n{a^{n – 1}}h + \frac{{n\left( {n – 1} \right)}}{{2!}}{a^{n – 2}}{h^2} + \cdots + {h^n} – {a^n}}}{h} \\ = \mathop {\lim }\limits_{h \to 0} \frac{{n{a^{n – 1}}h + \frac{{n\left( {n – 1} \right)}}{{2!}}{a^{n – 2}}{h^2} + \cdots + {h^n}}}{h} \\ = \mathop {\lim }\limits_{h \to 0} \frac{{h\left[ {n{a^{n – 1}} + \frac{{n\left( {n – 1} \right)}}{{2!}}{a^{n – 2}}h + \cdots + {h^{n – 1}}} \right]}}{h} \\ = \mathop {\lim }\limits_{h \to 0} \left( {n{a^{n – 1}} + \frac{{n\left( {n – 1} \right)}}{{2!}}{a^{n – 2}}h + \cdots + {h^{n – 1}}} \right) \\ \end{gathered}$

Applying the limits, we have
$\begin{gathered} \mathop {\lim }\limits_{x \to a} \frac{{{x^n} – {a^n}}}{{x – a}} = \left( {n{a^{n – 1}} + \frac{{n\left( {n – 1} \right)}}{{2!}}{a^{n – 2}}\left( 0 \right) + \cdots + {{\left( 0 \right)}^{n – 1}}} \right) \\ \Rightarrow \mathop {\lim }\limits_{x \to a} \frac{{{x^n} – {a^n}}}{{x – a}} = n{a^{n – 1}} + 0 + \cdots + 0 \\ \Rightarrow \mathop {\lim }\limits_{x \to a} \frac{{{x^n} – {a^n}}}{{x – a}} = n{a^{n – 1}} \\ \end{gathered}$