Write the value of lim x -1-2-3- n - n 2

Lim_x →∞1 + 2 + 3 + ⋯ + n/n^2 = lim_x →∞n/2[2 × 1 + (n 1) × 1]/n^2 = lim_x →∞n/2 (n +1)/n^2 = lim_x →∞n^2 (1 + 1/n )/2n^2 = lim_x →∞1 + 1/n/2 = lim_x →∞1 ...

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Standard XII

Mathematics

Expansions to Remove Indeterminate Form

Write the val...

Question

Write the value of ${lim}_{x \to \infty} \frac{1 + 2 + 3 + \dots + n}{n^{2}}$

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Solution

${lim}_{x \to \infty} \frac{1 + 2 + 3 + \dots + n}{n^{2}} = {lim}_{x \to \infty} \frac{\frac{n}{2} [2 \times 1 + (n - 1) \times 1]}{n^{2}} = {lim}_{x \to \infty} \frac{\frac{n}{2} (n + 1)}{n^{2}} = {lim}_{x \to \infty} \frac{n^{2} (1 + \frac{1}{n})}{2 n^{2}} = {lim}_{x \to \infty} \frac{1 + \frac{1}{n}}{2} = {lim}_{x \to \infty} \frac{1 + 1 + 3 + \dots + n}{n^{2}} = \frac{1}{2}$

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Write the value of limx→∞1+2+3+⋯+nn2

limx→∞1+2+3+⋯+nn2=limx→∞n2[2×1+(n−1)×1]n2=limx→∞n2(n+1)n2=limx→∞n2(1+1n)2n2=limx→∞1+1n2=limx→∞1+1+3+⋯+nn2=12

Write the value of ${lim}_{x \to \infty} \frac{1 + 2 + 3 + \dots + n}{n^{2}}$