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}