limit involving sine of fractional part

Question

What can you say about the following limit :

$$ \lim_{x\rightarrow 1} \dfrac{x\sin\lbrace x\rbrace}{x-1} $$

where $\lbrace x\rbrace$ is the fractional part of x

Whether this limit exists ?

Answer 1

Let's consider two cases, for $0 < x < 1$, we have $$ \frac{x\sin\{x\}}{x-1} = \frac{x\sin x}{x-1} $$ Now the numerator tends to $\sin 1$, the denominator to $0^-$, hence $$ \lim_{x \nearrow 1} \frac{x\sin\{x\}}{x-1} = -\infty $$ For $1 < x < 2$, $\{x\} = x-1$, hence $$ \frac{x\sin\{x\}}{x-1} = x\frac{\sin(x-1)}{x-1} \to 1 $$ Hence $$ \lim_{x \searrow 1} \frac{x\sin\{x\}}{x-1} = 1 $$ As the limits from both sides do not agree, the limit does not exist.