# The Value of Lim N → ∞ { 1 + 2 + 3 + . . . + N N + 2 − N 2 } - Mathematics

MCQ

The value of $\lim_{n \to \infty} \left\{ \frac{1 + 2 + 3 + . . . + n}{n + 2} - \frac{n}{2} \right\}$

• 1/2

• −1

• −1/2

#### Solution

−1/2

$\lim_{n \to \infty} \left[ \frac{1 + 2 + 3 + . . . . . n}{n + 2} - \frac{n}{2} \right]$
$= \lim_{n \to \infty} \left[ \frac{n\left( n + 1 \right)}{2\left( n + 2 \right)} - \frac{n}{2} \right]$
$= \lim_{n \to \infty} \frac{n}{2} \left[ \frac{n + 1 - n - 2}{n + 2} \right]$
$= \lim_{n \to \infty} \frac{n}{2}\left( \frac{- 1}{n + 2} \right)$
$= \lim_{n \to \infty} \frac{- 1}{2\left( 1 + \frac{2}{n} \right)}$
$= \frac{- 1}{2\left( 1 + 0 \right)}$
$= - \frac{1}{2}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Q 38 | Page 81