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Question Number 162726 by LEKOUMA last updated on 31/Dec/21

$$1)Calculate$$$$\underset{x\to 0}{\lim }\frac{{tgx}^m}{{\left(\sin x\right)}^n},(m,n\in N)$$$$2)f^{\prime }\left(a\right)e\mathrm{xiste},calculate$$$$\underset{x\to +\mathrm{\infty }}{\lim }x[f(a+\frac{a}{x})-f(a-\frac{\beta }{x})],$$$$(\alpha ,\beta \in R)$$

Answered by mr W last updated on 01/Jan/22

$$\left(1\right)$$$$\underset{x\to 0}{\lim }\frac{{tgx}^m}{{\left(\sin x\right)}^n}$$$$=\underset{x\to 0}{\lim }\frac{{tgx}^m}{x^m}\times {\left(\frac{x}{\sin x}\right)}^n\times x^{m-n}$$$$=\underset{x\to 0}{\lim }{x}^{m-n}=\{\begin{array}{l}1ifm=n\\ 0ifm>n\\ \mathrm{\infty }ifm<n\end{array}$$$$$$$$\left(2\right)$$$$\underset{x\to +\mathrm{\infty }}{\lim }x[f(a+\frac{\alpha }{x})-f\left(a\right)-f(a-\frac{\beta }{x})+f\left(a\right)]$$$$=\underset{x\to +\mathrm{\infty }}{\lim }[\frac{f(a+\frac{\alpha }{x})-f\left(a\right)}{\frac{1}{x}}-\frac{f(a-\frac{\beta }{x})-f\left(a\right)}{\frac{1}{x}}]$$$$=\underset{x\to +\mathrm{\infty }}{\lim }[\alpha \times \frac{f(a+\frac{\alpha }{x})-f\left(a\right)}{\frac{\alpha }{x}}+\beta \times \frac{f(a-\frac{\beta }{x})-f\left(a\right)}{-\frac{\beta }{x}}]$$$$=\underset{h,k\to 0}{\lim }[\alpha \times \frac{f(a+h)-f\left(a\right)}{h}+\beta \times \frac{f(a+k)-f\left(a\right)}{k}]$$$$=\alpha f^{\prime }\left(a\right)+\beta f^{\prime }\left(a\right)$$$$=(\alpha +\beta )f^{\prime }\left(a\right)$$

Commented by Ar Brandon last updated on 01/Jan/22

👏👏👏First comment of the year (GMT+1) Happy New year, Sir !

Commented by mr W last updated on 01/Jan/22

$$thanks!$$$$thesametoyouandallothers!$$

Commented by Ar Brandon last updated on 01/Jan/22

$$\mathrm{Thanks}\mathrm{Sir}.$$

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