Examples of Limit

Example:
If $$f\left( {\text{x}} \right) = {{\text{x}}^3} – 2{{\text{x}}^2} + 3{\text{x}} – 7$$,

then evaluate the limit $$\mathop {\lim }\limits_{{\text{x}} \to 2} f\left( {\text{x}} \right)$$.

Solution:
We have
\[f\left( {\text{x}} \right) = {{\text{x}}^3} – 2{{\text{x}}^2} + 3{\text{x}} – 7\]

\[\begin{gathered} \therefore \mathop {\lim }\limits_{{\text{x}} \to 2} f\left( {\text{x}} \right) = \mathop {\lim }\limits_{{\text{x}} \to 2} \left( {{{\text{x}}^3} – 2{{\text{x}}^2} + 3{\text{x}} – 7} \right) \\ \Rightarrow \mathop {\lim }\limits_{{\text{x}} \to 2} f\left( {\text{x}} \right) = \mathop {\lim }\limits_{{\text{x}} \to 2} \left( {{{\text{x}}^3}} \right) – \mathop {\lim }\limits_{{\text{x}} \to 2} \left( {2{{\text{x}}^2}} \right) + \mathop {\lim }\limits_{{\text{x}} \to 2} \left( {3{\text{x}}} \right) – \mathop {\lim }\limits_{{\text{x}} \to 2} \left( 7 \right) \\ \Rightarrow \mathop {\lim }\limits_{{\text{x}} \to 2} f\left( {\text{x}} \right) = {\left( {\text{2}} \right)^3} – 2{\left( {\text{2}} \right)^2} + 3\left( {\text{2}} \right) – 7 = 8 – 8 + 6 – 7\;\;\;\; = – 1 \\ \end{gathered} \]