Question

The value of $\underset{x\to 0}{\lim }\frac{sinx}{3}\left[\frac{5}{x}\right]$ 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)

$\frac{5}{x}-1<\left[\frac{5}{x}\right]\le \frac{5}{x}$
$\underset{h\left(x\right)}{\underbrace{\frac{sinx}{3}\left(\frac{5}{x}-1\right)}}<\underset{f\left(x\right)}{\underbrace{\frac{sinx}{3}\left[\frac{5}{x}\right]}}\le \underset{g\left(x\right)}{\underbrace{\frac{sinx}{3}\left(\frac{5}{x}\right)}}$
by sandwich theorem
$\because$ $\underset{x\to 0}{\lim }g\left(x\right)=\underset{x\to 0}{\lim }h\left(x\right)=\frac{5}{3}$
$\therefore \underset{x\to 0}{\lim }f\left(x\right)=\frac{5}{3}$