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Question Number 98768 by bemath last updated on 16/Jun/20

$$\underset{x\to 0}{\lim }\frac{\sqrt{\mathrm{x}+1}\sin \mathrm{x}+\ln (1+{\mathrm{x}}^2)-\mathrm{x}}{\sqrt[3]{1+{\mathrm{x}}^2}-1}$$

Answered by john santu last updated on 16/Jun/20

Commented by Farruxjano last updated on 16/Jun/20

$${\mathit{L}}^{\prime }{\mathit{H}\mathit{o}\mathit{s}\mathit{p}\mathit{i}\mathit{t}\mathit{a}\mathit{l}}^{\prime }\mathit{s}\mathit{r}\mathit{u}\mathit{l}\mathit{e}:$$$$=\underset{\mathit{x}\to 0}{\mathit{l}\mathit{i}\mathit{m}}\frac{{\mathit{x}}^2+1+2\mathit{x}-1}{\frac{2\mathit{x}}{3}}=\underset{\mathit{x}\to 0}{\mathit{l}\mathit{i}\mathit{m}}\frac{\mathit{x}+2}{\frac{2}{3}}=3$$

Commented by bemath last updated on 16/Jun/20

$$\mathrm{sorry}.\mathrm{wrong}\mathrm{sir}$$$$\frac{\mathrm{d}}{\mathrm{dx}}({\mathrm{x}}^2+2\mathrm{x})=2\mathrm{x}+2$$

Commented by john santu last updated on 16/Jun/20

$$\mathrm{your}\mathrm{answer}\mathrm{is}\mathrm{wrong}.\mathrm{please}\mathrm{check}$$

Commented by HamraboyevFarruxjon last updated on 16/Jun/20

$${\mathit{L}}^{\prime }{\mathit{H}\mathit{o}\mathit{p}\mathit{i}\mathit{t}\mathit{a}\mathit{l}}^{\prime }\mathit{s}\mathit{r}\mathit{u}\mathit{l}\mathit{e}:$$$$=\underset{\mathit{x}\to 0}{\mathit{l}\mathit{i}\mathit{m}}\frac{2\mathit{x}+1+2\mathit{x}-1}{\frac{2}{3}\mathit{x}}=6$$

Commented by mathmax by abdo last updated on 16/Jun/20

$${\lim }_{\mathrm{x}\to 0}\frac{\mathrm{x}(1+\frac{\mathrm{x}}{2})+{\mathrm{x}}^2-\mathrm{x}}{(1+\frac{{\mathrm{x}}^2}{3})-1}={\lim }_{\mathrm{x}\to 0}\frac{\frac{3}{2}{\mathrm{x}}^2}{\frac{{\mathrm{x}}^2}{3}}=\frac{9}{2}$$

Answered by bramlex last updated on 16/Jun/20

$${(1+x)}^n\sim (1+nx);x\to 0$$$$\sin x\sim x;x\to 0$$$$\ln (1+x)\sim x;x\to 0$$$$then\underset{x\to 0}{\lim }\frac{\sqrt{x+1}\sin x+\ln (1+x^2)-x}{\sqrt[3]{1+x^2}-1}$$$$=\underset{x\to 0}{\lim }\frac{(1+\frac{x}{2})x+x^2-x}{(1+\frac{x^2}{3})-1}$$$$=\underset{x\to 0}{\lim }\frac{\left(\frac{3x^2}{2}\right)}{\left(\frac{x^2}{3}\right)}=\frac{3}{2}\times 3=\frac{9}{2}.$$

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