Question

For each $t \in R$ , let [t] be the greatest integer less than or equal to t. Then $\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\left[ {\dfrac{1}{x}} \right] + \left[ {\dfrac{2}{x}} \right] + ..... + \left[ {\dfrac{{15}}{x}} \right] } \right)$ .
A is equal to 120
B Does not exist (in R)
C Is equal to 0
D Is equal to 15

Answer 1

Hint : A limit is defined as a value that a function approaches the output for the given input values and the limit of a constant times a function is equal to the constant times the limit of the function. As to find the limits with respect to the equation we can see that $x \to {0^ + }$ which implies that x is approaching in the positive direction to infinite, hence we can find the value by simplifying the terms for the real numbers.

Complete step-by-step answer :
Let us write the given data:
$\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\left[ {\dfrac{1}{x}} \right] + \left[ {\dfrac{2}{x}} \right] + ..... + \left[ {\dfrac{{15}}{x}} \right] } \right)$
As given, $x \to {0^ + }$
$\Rightarrow$ $\dfrac{1}{x} \to + \infty$
= \mathop {\lim }\limits_{x \to {0^ + }} x\left( {\dfrac{1}{x} + \dfrac{2}{x} + ... + \dfrac{{15}}{x}} \right) - x\left( {\left\\{ {\dfrac{1}{x}} \right\\} + \left\\{ {\dfrac{2}{x}} \right\\} + ....\left\\{ {\dfrac{{15}}{x}} \right\\}} \right)
The \left\\{ {} \right\\} denotes the fractional part of x, hence we get the equation as
= $\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\dfrac{1}{x} + \dfrac{2}{x} + ... + \dfrac{{15}}{x}} \right) - 0$
Since,
0 \leqslant \left\\{ {\dfrac{r}{x}} \right\\} < 1 and
0 \leqslant x\left\\{ {\dfrac{r}{x}} \right\\} < x
Hence, we get
= $\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\dfrac{{1 + 2 + ..... + 15}}{x}} \right)$
Simplifying the common terms i.e., x we get
= $1 + 2 + ......... + 15$
We know that, $\dfrac{{n\left( {n + 1} \right)}}{2}$
= $\dfrac{{15\left( {15 + 1} \right)}}{2}$
= $\dfrac{{15 \times 16}}{2}$
= 120
Hence, option A is the right answer.
So, the correct answer is “Option A”.

Note : In order for a limit to exist, the function has to approach a particular value. Since the function doesn't approach a particular value, the limit does not exist.
If a limit approaches positive infinity from one side, and negative infinity from the other side, it does not approach the same thing from both sides. This is why the limit doesn't exist. In order for a limit to exist it must approach the same thing from both sides.
A function may approach two different limits. One where the variable approaches its limit through values larger than the limit and the other where the variable approaches its limit through values smaller than the limit, for a function whose values are with infinite limits, the function diverges and the usual limit does not exist. However, in this case one may introduce limits with infinite values.