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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Find which of the function:
$\text{f(x)}=|\text{x}|+|\text{x}-1|\text{ at x}=1$
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Answer

The function f will be continuous at x = a, if $=\lim\limits_{\text{h}\rightarrow\text{a}^-}\text{f(x)}=\lim\limits_{\text{h}\rightarrow\text{a}^+}\text{f(x)}=\text{f(a)}.$
Consider, $\text{f(x)}=|\text{x}|+|\text{x}-1|\text{ at x}=1$
At x = 1, $\text{L.H.L}=\lim\limits_{\text{h}\rightarrow1^-}\big[|\text{x}|+|\text{x}-1|\big]$
$=\lim\limits_{\text{h}\rightarrow0}\big[|1-\text{h}|+|1-\text{h}-1|\big]=1+0=1$
At x = 1, $\text{R.H.L}=\lim\limits_{\text{h}\rightarrow1^-}\big[|\text{x}|+|\text{x}-1|\big]$
$=\lim\limits_{\text{h}\rightarrow0}\big[|1+\text{h}|+|1+\text{h}-1|\big]=1+0=1$
$\text{f}(1)=|1|+|0|=1$
Since, L.H.L = R.H.L = f(1)
Hence, f(x) is continuous at x = 1.

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