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

The function is continuous at $x = 1$

The correct answer is Option (1) → The function is continuous at $x = 1$ ##

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

Put $x = 1 + h$

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

Put $x = 1 - h$

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

$∴\lim\limits_{x \to 1^+} f(x) = \lim\limits_{x \to 1^-} f(x) = f(1) \quad \dots(i)$

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

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