If f(x)= l 2x+b(x< ) +d( x ) . is such that _ x a f(x=L), then L. - Vidyadip

MCQ

If $ \mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}2\mathrm{x}+\mathrm{b}(\mathrm{x}<\alpha)\\\mathrm{x}+\mathrm{d}(\mathrm{\text{x}}\geq\alpha)\end{array}\right.$ is such that $ \lim_\limits{\text{x} \rightarrow \text{a}}\text{f}(\text{x}=\text{L}),$ then $L.$

✓
$2d - b$
B
$b - db$
C
$d + bd$
D
$b- 2d$

✓

Answer

Correct option: A.

$2d - b$

$\lim _{h \rightarrow 0^{-}} f(x)=2(\alpha-h)+b=2 \alpha+b=L \ldots \ldots \ldots \ldots(1) .$
$\lim _{\mathrm{h} \rightarrow 0^{+}} \mathrm{f}(\mathrm{x})=(\alpha+\mathrm{h})+\mathrm{d}=\mathrm{L} \alpha=\mathrm{L}-\mathrm{d} \ldots \ldots \ldots \ldots \ldots (2).$
Substituting value of euation $(2)$ in $(1),$ we get
$2(\mathrm{~L}-\mathrm{d})+\mathrm{b}=\mathrm{L}$
$ \mathrm{~L}=2 \mathrm{~d}-\mathrm{b}$
Hence, option $A$ is correct.

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