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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
MCQ
If $f(x)=\left\{\begin{array}{ll}\frac{x-2}{|x-2|}+a, & x<2 \\ a+b, & x=2 \\ \frac{x-2}{|x-2|}+b, & x>2\end{array}\right.$ is continuous at $x=2$, then $a + b =$
  • A
    2
  • B
    1
  • ✓
    $0$
  • D
    -1
✓

Answer

Correct option: C.
$0$
(C)
$\lim _{x \rightarrow 2^{-}} f (x)=\lim _{ h \rightarrow 0} f (2- h )$
$=\lim _{h \rightarrow 0} \frac{2-h-2}{|2-h-2|}+a$
$=\lim _{h \rightarrow 0}\left(-\frac{h}{h}+a\right)=a-1$
$\lim _{x \rightarrow 2^{+}} f(x)=\lim _{h \rightarrow 0} f(2+h)$
$=\lim _{h \rightarrow 0} \frac{2+h-2}{|2+h-2|}+b=b+1$ and $f(2)=a+b$
Since $f (x)$ is continuous at $x=2$,
$\therefore \quad \lim _{x \rightarrow 2^{-}} f (x)= f (2)=\lim _{x \rightarrow 2^{+}} f (x)$
$\Rightarrow a -1= a + b = b +1$
$\Rightarrow b =-1$ and $a =1$

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