For each t∈R, let [t] be the greatest integer less than or equal to t. Then,limx→0+ x1x+2x+……+15x

A
is equal to 15
B
is equal to 120
C
does not exist in R
D
is equal to 0.

Solution:

We know that $[x] = x - {x}$ , where {x} denotes the fractional part of x.

$\begin{aligned} ∴ lim_{x \to 0^{+}} x [(\frac{1}{x} + \frac{2}{x} + \dots + \frac{15}{x}) - (\{\frac{1}{x}\} + \{\frac{2}{x}\} + \dots + \{\frac{15}{x}\})] \\ = lim_{x \to 0^{+}} (1 + 2 + \dots + 15) - lim_{x \to 0^{+}} x (\{\frac{1}{x}\} + \{\frac{2}{x}\} + \dots + \{\frac{15}{x}\}) \end{aligned}$

$= \frac{15 (15 + 1)}{2} + 0 \times A$ finite number less than 15=120

For each $t \in R$ , let [t] be the greatest integer less than or equal to t. Then,
$lim_{x \to 0^{+}} x ([\frac{1}{x}] + [\frac{2}{x}] + \dots \dots + [\frac{15}{x}])$

Solution:

Related content