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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability3 Marks
Question
Examine the continuity of function.$
f(x)=\left\{\begin{array}{ll}
1+x, & x \leq 3 \\
7-x, & x>3
\end{array} \text { at } x=3\right.
$
✓

Answer

ATQ $\quad f(x)=\left\{\begin{array}{ll}1+x, & x \leq 3 \\ 7-x, & x>3\end{array}\right.$
at $x=3 \quad f(3)=1+3=4 ...(1)$$
\begin{aligned}
\text { R.H.L. } f(3+0) & =\lim _{h \rightarrow 0} f(3+h) \\
& =\lim _{h \rightarrow 0} 7-(3-h)=\lim _{h \rightarrow 0}(4-h)=4.....(2)
\end{aligned}
$
L.H.L. $f(3-0)=\lim _{h \rightarrow 0} f(3-h)$$
=\lim _{h \rightarrow 0}(1+3-h)=\lim _{h \rightarrow 0}(4-h)=4.....(3)
$
$
\because \quad f(3)=f(3+0)=f(3-0)=4
$
$\therefore$ function is continuous at $x=3$.

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