First slide

Introduction to limits

Question

$lim_{x \to 2^{+}} \{\frac{[x]^{3}}{3} - {[\frac{x}{3}]}^{3}\}$ is equal to

Moderate

A

0
B

$\frac{64}{27}$
C

$\frac{8}{3}$
D

none of these

Solution

$lim_{x \to 2^{+}} \{\frac{[x]^{3}}{3} - {[\frac{x}{3}]}^{3}\}$
$\begin{array}{l} = \frac{1}{3} lim_{x \to 2^{+}} [x]^{3} - lim_{x \to 2^{+}} {[\frac{x}{3}]}^{3} \\ = \frac{1}{3} \times 2^{3} - 0 = \frac{8}{3} \end{array}$

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