Question

Examinie the continuity of the function $f\left(x\right)=x^3+2x^2-1$ at $x=1$

Answer 1

We have, $f\left(x\right)=x^3+2x^2-1atx=1\therefore \underset{x\to 1^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}(1-h{)}^3+2(1-h{)}^2-1=2$
and $\underset{x\to 1^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}(1+h{)}^3+2(1+h{)}^2-1=2\therefore \underset{x\to 1^{+}}{lim}f\left(x\right)=\underset{x\to 1^{-}}{lim}f\left(x\right)$
and $f\left(1\right)=1+2-1=2$
So, f(x) is continuous at x=1,
Note Every polynomial function is continuous at any real point.