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

Question

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

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Answer

We know that, function f will be continuous at x = a, if $\lim\limits_{\text{x}\rightarrow\text{a}^-}\text{f(x)}=\lim\limits_{\text{x}\rightarrow\text{a}^+}\text{f(x)}=\text{f(a)}.$
Consider, $f(x) = x^3 + 2x^2 - 1 $at $x = 1.$
$\lim\limits_{\text{x}\rightarrow1^+}\text{f(x)}=\lim\limits_{\text{h}\rightarrow0}(1+\text{h})^3+2(1+\text{h})^2-1=2$
and
$\lim\limits_{\text{x}\rightarrow1^-}\text{f(x)}=\lim\limits_{\text{h}\rightarrow0}(1-\text{h})^3+2(1-\text{h})^2-1=2$
$\because\ \lim\limits_{\text{x}\rightarrow1^+}\text{f(x)}=\lim\limits_{\text{x}\rightarrow1^-}\text{f(x)}$
And $f(1) = 1 + 2 - 1 = 2$
Thus, f(x) is continuous at $x = 1.$

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