Limit of Functions Involving nth Power

In this tutorial we shall discuss an example to evaluating limits involving function with nth power of variable, in most of the cases if limit involves nth power variable expression and to solve using 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}