Question

The value of $\underset{x \rightarrow 0}{l i m}\frac{sin x}{3}\left[\frac{5}{x}\right]$ is equal to

(where, $\left[.\right]$ represents the greatest integer function)

• A. $\frac{1}{3}$
• B. $0$
• C. $\frac{5}{3}$
• D. $1$

Answer 1

Answer: (C)

$\frac{5}{x}-1 < \left[\frac{5}{x}\right]\leq \frac{5}{x}$
$\underset{h \left(x\right)}{\underbrace{\frac{sin x}{3} \left(\frac{5}{x} - 1\right)}} < \underset{f \left(x\right)}{\underbrace{\frac{sin ⁡ x}{3} \left[\frac{5}{x}\right]}}\leq \underset{g \left(x\right)}{\underbrace{\frac{sin ⁡ x}{3} \left(\frac{5}{x}\right)}}$
by sandwich theorem
$\because $ $\underset{x \rightarrow 0}{l i m}g\left(x\right)=\underset{x \rightarrow 0}{l i m}h\left(x\right)=\frac{5}{3}$
$\therefore \underset{x \rightarrow 0}{l i m}f\left(x\right)=\frac{5}{3}$