Let fx- xsin x x-1- Then where x denotes fractional part of x

Given : f(x)=xsin{ x }x 1 R.H.L.=lim_h→ 0(1+h)sin{ 1+h }h =lim_h→ 0(1+h)sin hh=1 L.H.L.=lim_h→ 0(1 h)sin{ 1 h } h =lim_h→ 0(1 h)sin(1 h ) h= ∞ ∴ f(x) is discont ...

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Byju's Answer

Standard XII

Mathematics

Algebra of Limits

Let fx= xsin ...

Question

Let $f (x) = \frac{x sin {x}}{x - 1}$ . Then
( where ${x}$ denotes fractional part of $x$ )

f (x)

is continuous at

x = 1

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f (x)

is discontinuous at

x = 1

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lim x \to 1^{-} f (x) = 1

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lim x \to 1^{+} f (x) = - 1

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Solution

The correct option is B $f (x)$ is discontinuous at $x = 1$
Given : $f (x) = \frac{x sin {x}}{x - 1}$
$R . H . L . = lim h \to 0 \frac{(1 + h) sin {1 + h}}{h}$
$= lim h \to 0 \frac{(1 + h) sin h}{h} = 1$
$L . H . L . = lim h \to 0 \frac{(1 - h) sin {1 - h}}{- h}$
$= lim h \to 0 \frac{(1 - h) sin (1 - h)}{- h} = - \infty$
$∴$ $f (x)$ is discontinuous at $x = 1$

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The correct option is B f(x) is discontinuous at x=1Given : f(x)=xsin{x}x−1 R.H.L.=limh→0(1+h)sin{1+h}h =limh→0(1+h)sinhh=1 L.H.L.=limh→0(1−h)sin{1−h}−h =limh→0(1−h)sin(1−h)−h=−∞ ∴ f(x) is discontinuous at x=1

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