First slide

Introduction to limits

Question

If {x} denotes tile part of x, then $lim_{x \to 1} \frac{x \sin {x}}{x - 1}$ is

Moderate

A

0
B

-1
C

non-existent
D

none of these

Solution

We have,
$lim_{x \to 1^{-}} \frac{x \sin {x}}{x - 1} = lim_{x \to 1^{-}} \frac{x \sin x}{x - 1} \to - \infty$
and,
$lim_{x \to 1^{+}} \frac{x \sin {x}}{x - 1} = lim_{x \to 1^{+}} \frac{x \sin (x - 1)}{x - 1} = 1 \times 1 = 1$
Clearly, $lim_{x \to 1^{-}} \frac{x \sin {x}}{x - 1} \neq lim_{x \to 1^{+}} \frac{x \sin {x}}{x - 1}$
So $lim_{x \to 1} \frac{x \sin {x}}{x - 1}$ does not exist.

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