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CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY3 Marks
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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