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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 :
$f(x)=4+\sin x$, for $x<\pi=3-\cos x$ for $x>\pi$.
✓

Answer

$f(x)=4+\sin x, x<\pi=3-\cos x, x>\pi$
$\sin x$ and $\cos x$ are continuous for all $x \in R$.
4 and 3 are constant functions.
$\therefore 4+\sin x$ and $3-\cos x$ are continuous for all $x \in R$.
$\therefore f ( x )$ is continuous for both the given intervals.
Let us test the continuity at $x=\pi$.
$ \lim _{x \rightarrow \pi^{-}} f(x)=\lim _{x \rightarrow x^{-}}(4+\sin x)$
$=4+\sin \pi$
$=4+0$
$=4$
$\lim _{x \rightarrow x ^{+}} f (x)=\lim _{x \rightarrow x ^{+}}(3-\cos x)$
$=3-\cos \pi$
$=3-(-1)$
$=4$
$\therefore \quad \lim _{x \rightarrow s ^{-}} f (x)=\lim _{x \rightarrow z ^{+}} f (x)$
$\therefore \quad \lim _{x \rightarrow \pi} f(x)=4$
But $f(\pi)$ is not defined.
$\therefore f ( x )$ has a removable discontinuity at $x =\pi$.

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