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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity3 Marks
Question
Discuss the continuity of the function $f(x)=|2 x+3|$, at $x=\frac{-3}{2}$.
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Answer

$f(x)=|2 x+3|, x=\frac{-3}{2}$
$|2 x+3|=2 x+3 \quad ; x \geq \frac{-3}{2}$
$=-(2 x+3) ; x<\frac{-3}{2}$
$\lim _{x \rightarrow \frac{3^{-}}{2}} f(x)=\lim _{x \rightarrow \frac{-3^{-}}{2}}|2 x+3|$
$=\lim _{x \rightarrow \frac{-3^{-}}{2}}[-(2 x+3)]$
$=-\left[2\left(\frac{-3}{2}\right)+3\right]$
$=0$
$\lim _{x \rightarrow \frac{-3^{+}}{2}} f(x)=\lim _{x \rightarrow \frac{-3^{+}}{2}}|2 x+3|$
$=\lim _{x \rightarrow \frac{3^{+}}{2}}(2 x+3)$
$=2\left(\frac{-3}{2}\right)+3$
$=0$
$f\left(\frac{-3}{2}\right)=\left|2\left(\frac{-3}{2}\right)+3\right|$
$=|0|$
$\text { = } 0$
$\therefore \quad \lim _{x \rightarrow \frac{-3^{-}}{2}} f (x)=\lim _{x \rightarrow \frac{-3^{+}}{2}} f (x)= f \left(\frac{-3}{2}\right)$
$\therefore \quad f (x) \text { is continuous at } x=\frac{-3}{2} \text {. }$

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