Question

\( \lim _{x \rightarrow \infty}\left(\frac{1}{x^{2}}+\frac{2}{x^{2}}+\ldots .+\frac{x}{x^{2}}\right) \)

Answer 1

Since there's a common denominator of x² , the numerator can be added up using the sum of natural numbers formula (\(1+2+3+...+n=\) \((n)(n+1)\over2\))

Hence the limit becomes

\(\lim\limits_{x \to \infty} \frac{(x)(x+1)}{2x^2}\)

Cancelling the x from the numerator, we get

\(\lim\limits_{x \to \infty} \frac{(x+1)}{2x}\)

Using sum law of limits,

\(\lim\limits_{x \to \infty} \frac{(x+1)}{2x} = \lim\limits_{x \to \infty}\frac{1}{2} + \lim\limits_{x \to \infty} \frac{1}{2x}\)

\(= {1\over2} \)