Determine the continuity of $f (x) = x^{3} + 2 x^{2} - x + 4$ .

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Solution

The function $f (x) = x^{3} + 2 x^{2} - x + 4$ is defined at every real number $c$ and it's value at $c$ is $f (c) = c^{3} + 2 c^{2} - c + 4$ .
We also know that,
$lim x \to c f (x) = lim x \to c (x^{3} + 2 x^{2} - x + 4) = c^{3} + 2 c^{2} - c + 4.$
Thus, $lim x \to c f (x) = f (c)$ .
Hence, $f (x)$ is continuous at every real number.
This means $f (x) = x^{3} + 2 x^{2} - x + 4$ is a continuous function.

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