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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
Question
Identify discontinuities for the following functions as either a jump or a removable discontinuity :
$f(x)=x^2+3 x-2$, for $x \leq 4$
$=5 x+3$, for $x>4$.
✓

Answer

$ f(x)=x^2+3 x-2, x \leq 4$
$=5 x+3, x>4 $
$f(x)$ is a polynomial function for both the intervals.
$\therefore f ( x )$ is continuous for both the given intervals.
Let us test the continuity at $x =4$.
$ \lim _{x \rightarrow 4^{-}} f(x)=\lim _{x \rightarrow 4^{-}}\left(x^2+3 x-2\right)$
$=(4)^2+3(4)-2$
$=26$
$\lim _{x \rightarrow 4^{+}} f (x)=\lim _{x \rightarrow 4^{+}}(5 x+3)$
$=5(4)+3$
$=23$
$\therefore \quad \lim _{x \rightarrow 4^{-}} f (x) \neq \lim _{x \rightarrow 4^{+}} f (x)$
$\therefore \quad \lim _{x \rightarrow 4} f (x) \text { does not exist. }$
$\therefore f ( x )$ is discontinuous at $x =4$.
$\therefore f ( x )$ has a jump discontinuity at $x =4$.

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