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Question Number 160672 by LEKOUMA last updated on 04/Dec/21

$$Calculate$$$$\underset{x\to 0}{\lim }\frac{{tgx}^m}{{\left(\sin x\right)}^n},$$$$\underset{x\to 0}{\lim }\frac{x\cos x-x}{(e^x-1)\ln (1+3x^2)}$$

Commented by tounghoungko last updated on 04/Dec/21

$$\left(2\right)\underset{x\to 0}{\lim }\frac{\mathrm{x}\cos \mathrm{x}-\mathrm{x}}{({\mathrm{e}}^{\mathrm{x}}-1)\ln (1+{3\mathrm{x}}^2)}$$$$=\underset{x\to 0}{\lim }\frac{\mathrm{x}(\cos \mathrm{x}-1)}{({\mathrm{e}}^{\mathrm{x}}-1)\ln (1+{3\mathrm{x}}^2)}$$$$=\underset{x\to 0}{\lim }\frac{1}{\left(\frac{{\mathrm{e}}^{\mathrm{x}}-1}{\mathrm{x}}\right)}\times \underset{x\to 0}{\lim }\frac{-2\sin {}^2\left(\frac{1}{2}\mathrm{x}\right)}{\ln (1+{3\mathrm{x}}^2)}$$$$=1\times \underset{x\to 0}{\lim }\frac{-\frac{1}{2}{\mathrm{x}}^2}{\ln (1+{3\mathrm{x}}^2)}$$$$=-\frac{1}{2}\times \underset{x\to 0}{\lim }\frac{2\mathrm{x}}{\left(\frac{6\mathrm{x}}{(1+{3\mathrm{x}}^2)}\right)}$$$$=-\frac{1}{6}\times \underset{x\to 0}{\lim }(1+{3\mathrm{x}}^2)=-\frac{1}{6}$$

Answered by alephzero last updated on 04/Dec/21

$$1.\underset{x\to 0}{\lim }\frac{\mathrm{tg}\left(x^m\right)}{{\left(\sin x\right)}^n}=?$$$$n^0=1\iff n\in \mathbb{R}$$$$\mathrm{tg}0=0$$$$\mathrm{tg}1\approx 1.55741$$$$\Rightarrow \mathrm{tg}{0}^m=0\iff m>0$$$$\mathrm{tg}{0}^m\approx 1.55741\iff m=0$$$$\sin 0=0$$$$\Rightarrow {\left(\sin 0\right)}^n=0\iff n>0$$$${\left(\sin 0\right)}^n=1\iff n=0$$$$\Rightarrow \underset{x\to 0}{\lim }\frac{\mathrm{tg}{x}^m}{{\left(\sin x\right)}^n}=\pm \mathrm{\infty }\iff ((m>0)\wedge (n>0))$$$$\underset{x\to 0}{\lim }\frac{\mathrm{tg}{x}^m}{{\left(\sin x\right)}^n}=0\iff ((m>0)\wedge (n=0))$$$$\underset{x\to 0}{\lim }\frac{\mathrm{tg}{x}^m}{{\left(\sin x\right)}^n}\approx 1.55741\iff ((m=0)\wedge (n=0))$$$$Answeredbyalephzero,student$$$$of{6}^{th}grade.$$

Commented by mr W last updated on 04/Dec/21

$$inthisforumallmembersareequal.$$$$one{doesn}^{\prime }tneedtotellifhe/sheis$$$$teacherorstudent,oldoryoung,man$$$$orwoman,IndianorJapanese,$$$$buddistormuslim,earthlingor$$$$marsian...onlyonethingis$$$$important:theylovemathematics.$$

Commented by greg_ed last updated on 05/Dec/21

$$\mathrm{right}!$$

Answered by mr W last updated on 04/Dec/21

$$\left(1\right)$$$$=\underset{x\to 0}{\lim }\frac{\tan \left(x^m\right)}{x^m}\times {\left(\frac{x}{\sin x}\right)}^n\times x^{m-n}$$$$=\underset{x\to 0}{\lim }{x}^{m-n}=\{\begin{array}{l}1,ifm=n\\ 0,ifm>n\\ \mathrm{\infty },ifm<n\end{array}$$$$\left(2\right)$$$$=\underset{x\to 0}{\lim }\frac{x(-2{\sin }^2\frac{x}{2})}{(e^x-1)\ln (1+3x^2)}$$$$=\underset{x\to 0}{\lim }\frac{-2x{\sin }^2\frac{x}{2}}{(x+\frac{x^2}{2!}+...)(3x^2-\frac{9x^4}{2}+...)}$$$$=\underset{x\to 0}{\lim }\frac{-2{\sin }^2\frac{x}{2}}{(1+\frac{x}{2!}+...)(3-\frac{9x^2}{2}+...)x^2}$$$$=\underset{x\to 0}{\lim }\frac{-1}{(1+\frac{x}{2!}+...)(3-\frac{9x^2}{2}+...)\times 2}\times {\left(\frac{\sin \frac{x}{2}}{\frac{x}{2}}\right)}^2$$$$=-\frac{1}{1\times 3\times 2}\times 1^2$$$$=-\frac{1}{6}$$

Answered by Jamshidbek last updated on 05/Dec/21

$$\underset{x\to 0}{\lim }\frac{{\mathrm{tgx}}^{\mathrm{m}}}{{\left(\mathrm{sinx}\right)}^{\mathrm{n}}}=\underset{x\to 0}{\lim }\frac{{\mathrm{tgx}}^{\mathrm{m}}}{{\mathrm{x}}^{\mathrm{m}}}\cdot \frac{{\mathrm{x}}^{\mathrm{m}}}{{\left(\mathrm{sinx}\right)}^{\mathrm{n}}}=$$$$=\underset{x\to 0}{\lim }\frac{{\mathrm{x}}^{\mathrm{m}}\cdot {\mathrm{x}}^{\mathrm{m}}}{{\left(\frac{\mathrm{sinx}}{\mathrm{x}}\right)}^{\mathrm{n}}\cdot {\mathrm{x}}^{\mathrm{n}}}=\underset{x\to 0}{\lim }\frac{{\mathrm{x}}^{2\mathrm{m}}}{{\mathrm{x}}^{\mathrm{n}}}={\underset{x\to 0}{\lim x}}^{2\mathrm{m}-\mathrm{n}}=0$$

Commented by mr W last updated on 05/Dec/21

$$wrong!$$$$\lim \frac{{\mathrm{tgx}}^{\mathrm{m}}}{{\mathrm{x}}^{\mathrm{m}}}\cdot \frac{{\mathrm{x}}^{\mathrm{m}}}{{\left(\mathrm{sinx}\right)}^{\mathrm{n}}}\ne \lim \frac{{\mathrm{x}}^{\mathrm{m}}\cdot {\mathrm{x}}^{\mathrm{m}}}{{\left(\frac{\mathrm{sinx}}{\mathrm{x}}\right)}^{\mathrm{n}}\cdot {\mathrm{x}}^{\mathrm{n}}}$$$$x^{m}istoomuch.$$

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