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Question Number 216144 by klipto last updated on 28/Jan/25

$$1.{\mathbf{Lim}}_{\mathrm{n}\to \mathrm{\infty }}[\frac{1}{{\mathbf{n}}^2}+\frac{2}{{\mathbf{n}}^2}+\frac{3}{{\mathbf{n}}^2}+...+\frac{\mathbf{n}+1}{{\mathbf{n}}^2}]$$$$2.{\mathbf{lim}}_{\mathbf{x}\to 0}{\left(\frac{3\mathbf{sin}5\mathbf{x}}{\mathbf{x}}\right)}^{\frac{1-\mathbf{cos}4\mathbf{x}}{{\mathbf{x}}^2}}$$

Answered by Ghisom last updated on 28/Jan/25

$$\underset{n\to \mathrm{\infty }}{\lim }\underset{k=1}{\sum ^{n+1}}\frac{k}{n^2}=\underset{n\to \mathrm{\infty }}{\lim }\frac{\underset{k=1}{\sum ^{n+1}}k}{n^2}=\underset{n\to \mathrm{\infty }}{\lim }\frac{n^2+3n+2}{2n^2}=\frac{1}{2}$$

Answered by mr W last updated on 28/Jan/25

$$1)$$$$\underset{n\to \mathrm{\infty }}{\lim }\underset{k=1}{\sum ^{n+1}}\frac{k}{n^2}$$$$={\int }_0^{1}xdx$$$$=\frac{1}{2}$$

Commented by klipto last updated on 28/Jan/25

$$\mathbf{thanks}\mathbf{bossman}$$

Answered by mr W last updated on 28/Jan/25

$$2)$$$$=\underset{x\to 0}{\lim }{\left(\frac{15\sin 5x}{5x}\right)}^{\frac{2{\sin }^{2}2x}{x^2}}$$$$=\underset{x\to 0}{\lim }{\left(\frac{15\sin 5x}{5x}\right)}^{8{\left(\frac{\sin 2x}{2x}\right)}^2}$$$$={\left(15\times 1\right)}^{8\times 1^2}$$$$={15}^8$$

Commented by klipto last updated on 28/Jan/25

$$\mathbf{thanks}\mathbf{ghisom}$$

Commented by Ghisom last updated on 28/Jan/25

$$\mathrm{yes}.\mathrm{I}\mathrm{missed}\mathrm{the}\mathrm{power}...$$

Answered by MrGaster last updated on 29/Jan/25

$$\left(1\right):=\underset{n\to \mathrm{\infty }}{\lim }\left[\frac{1+2+3+\dots +(n+1)}{n^2}\right]$$$$=\underset{n\to \mathrm{\infty }}{\lim }\left[\frac{(n+1)(n+2)}{\frac{2}{n^2}}\right]$$$$=\underset{n\to \mathrm{\infty }}{\lim }\left[\frac{n^2+3n+2}{2n^2}\right]$$$$=\frac{1}{2}$$$$\left(2\right):=\underset{x\to 0}{\lim }{(3\cdot \frac{\sin 5x}{5x}\cdot 5)}^{\frac{1-(1-\frac{\left(4x^2\right)}{2!}+\dots )}{x^2}}$$$$=\underset{x\to 0}{\lim }{(15\cdot \frac{\sin 5x}{5x})}^{\frac{8x^2}{x^2}}$$$$={\underset{x\to 0}{\lim 15}}^8(\underset{x\to 0}{\lim }\frac{\sin 5x}{5x}=1)$$$$={15}^8$$

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