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Question Number 67031 by hmamarques1994@gmai.com last updated on 21/Aug/19

$$$$$$\underset{\mathbf{x}\to -1}{\mathbf{lim}}\sqrt[3]{\frac{\sqrt[7]{{\mathbf{x}}^5}+1}{1+\sqrt[9]{{\mathbf{x}}^7}}}=?$$$$$$

Commented by Tony Lin last updated on 22/Aug/19

$${\left(\underset{x\to -1}{\lim }\frac{\frac{5}{7}{x}^{\frac{-2}{7}}}{\frac{7}{9}{x}^{\frac{-2}{9}}}\right)}^{\frac{1}{3}}$$$$={\left(\underset{x\to -1}{\lim }\frac{{45}_9\sqrt{x^2}}{{49}_7\sqrt{x^2}}\right)}^{\frac{1}{3}}$$$$={\left(\frac{45}{49}\right)}^{\frac{1}{3}}$$$$pleasecheck$$

Commented by hmamarques1994@gmail.com last updated on 22/Aug/19

$$\mathit{y}\mathit{e}\mathit{s}\underset{}{!}$$$$=\frac{\sqrt[3]{315}}{7}\approx 0,972013$$$$\mathit{t}\mathit{h}\mathit{a}\mathit{n}\mathit{k}\mathit{y}\mathit{u}!$$

Commented by mathmax by abdo last updated on 22/Aug/19

$$youhaveused{lim}_{x\to x_0}{\left(\frac{f\left(x\right)}{g\left(x\right)}\right)}^r={\left({lim}_{x\to x_0}\frac{f^{{}^{\prime }}\left(x\right)}{g^{{}^{\prime }}\left(x\right)}\right)}^r$$$$isthiscorrect?(itsnothospitaltheorem....)$$

Commented by Tony Lin last updated on 22/Aug/19

$$L^{\prime }Hospitalrule:$$$$if\underset{x\to x_0}{\lim }f\left(x\right)=\underset{x\to x_0}{\lim }g\left(x\right)=0and\underset{x\to x_0}{\lim }\frac{f^{\prime }\left(x\right)}{g^{\prime }\left(x\right)}exists$$$$\Rightarrow \underset{x\to x_0}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to x_0}{\lim }\frac{f^{\prime }\left(x\right)}{g^{\prime }\left(x\right)}$$$$if\underset{x\to x_0}{\lim }f\left(x\right)=\pm \mathrm{\infty },\underset{x\to x_0}{\lim }g\left(x\right)=\pm \mathrm{\infty }and\underset{x\to x_0}{\lim }\frac{f^{\prime }\left(x\right)}{g^{\prime }\left(x\right)}exists$$$$\Rightarrow \underset{x\to x_0}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to x_0}{\lim }\frac{f^{\prime }\left(x\right)}{g^{\prime }\left(x\right)}$$$$if\underset{x\to x_0}{\lim }f\left(x\right)exists$$$$\underset{x\to x_0}{\lim }{\left[f\left(x\right)\right]}^n={\left[\underset{x\to x_0}{\lim }f\left(x\right)\right]}^n$$$$\Rightarrow PowerLawofLimit$$$$orwecanuseCompositionLawtoknow$$$$iffiscontinuousat\underset{x\to x_0}{\lim }g\left(x\right)$$$$then\underset{x\to x_0}{\lim }f\left(g\left(x\right)\right)=f\left(\underset{x\to x_0}{\lim }g\left(x\right)\right)$$

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