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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity3 Marks
Question
Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:
$
\begin{aligned}
& \mathrm{f}(\mathrm{x})=x^2+5 x+1, \text { for } 0 \leq x \leq 3 \\
& =x^3+x+5, \quad \text { for } 3<x \leq 6 \\
\end{aligned}
$
✓

Answer

$\begin{aligned} & \lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3^{-}}\left(x^2+5 x+1\right) \\ & =\lim _{x \rightarrow 3^{-}}(3)^2+5(3)+1 \\ & =9+15+1 \\ & =25 \\ & \lim _{x \rightarrow 3^{+}} f(x)=\lim _{x \rightarrow 3^{+}}\left(x^3+x+5\right) \\ & =(3)^3+3+5 \\ & =35 \\ & \therefore \lim _{x \rightarrow 3^{-}} f(x) \neq \lim _{x \rightarrow 3^{+}} f(x) \\ & \therefore \lim _{x \rightarrow 3} f(x) \text { does not exist } \\ & \therefore \mathrm{f}(x) \text { is discontinuous at } x=3 \\ & \therefore \mathrm{f}(x) \text { has a jump discontinuity at } x=3\end{aligned}$

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