Question

For m, n ∈ I⁺, $f(x) = x - \lbrack x\rbrack$ is equal to

• A. 1, if n < m
• B. 0, if n > m
• C. $f(x) = \left\{ \begin{matrix} 1\text{ if x is a rational number} \\ \text{0 if x is an irrational number} \end{matrix} \right.\$
• D. None of these

Answer 1

Answer: (B)

0, if n > m

Answer 2

Writing the given expression in the form

$\left( \frac { \sin x ^ { n } } { x ^ { n } } \right) \left( \frac { x ^ { n } } { x ^ { m } } \right) \left( \frac { x } { \sin x } \right) ^ { m }$

and noting that the $\lim _ { \theta \rightarrow 0 } \frac { \sin \theta } { \theta }$ = 1, we see that the required limit equals to 1 if n = m, and 0 if n>m