Examine the continuity of the function $f(x) = {x^3} + 2{x^2} - 1$ at $x = 1$.

We have, $f(x) = {x^3} + 2{x^2} - 1$ at $x = 1$

As we know that, function $f$ will be continuous at $x = a$, if $\mathop {\lim }\limits_{x \to {a^ - }} f(x) = \mathop {\lim }\limits_{x \to {a^ + }} f(x) = f(a)$.

therefore, $\mathop {\lim }\limits_{x \to {1^ + }} f(x) = \mathop {\lim }\limits_{h \to 0} {(1 + h)^3} + 2{(1 + h)^2} - 1 = 2$

and $\mathop {\lim }\limits_{x \to {1^ - }} f(x) = \mathop {\lim }\limits_{h \to 0} {(1 - h)^3} + 2{(1 - h)^2} - 1 = 2$
therefore, $\mathop {\lim }\limits_{x \to {1^ + }} f(x) = \mathop {\lim }\limits_{x \to {1^ - }} f(x)$

and $f(1) = 1 + 2 - 1 = 2$

So, $f(x)$ is continuous at $x = 1$.

Tip: Every polynomial function is continuous at every real point.