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Question Number 165392 by LEKOUMA last updated on 31/Jan/22

$$\underset{x\to +\mathrm{\infty }}{\lim }\frac{e^{\frac{1}{x}}-\cos \frac{1}{x}}{1-\sqrt{1-\frac{1}{x^2}}}$$$$\underset{x\to a}{\lim }\frac{x^x-a^a}{x-a}$$

Answered by Mathspace last updated on 31/Jan/22

$$f\left(x\right)=\frac{e^{\frac{1}{x}}-cos\left(\frac{1}{x}\right)}{1-\sqrt{1-\frac{1}{x^2}}}$$$$chamngement\frac{1}{x}=tgive$$$$f\left(x\right)=f\left(\frac{1}{t}\right)=\frac{e^t-cost}{1+\sqrt{1-t^2}}$$$$\sim \frac{1+t-(1-\frac{t^2}{2})}{1-(1-\frac{t^2}{2})}=\frac{t+\frac{t^2}{2}}{\frac{t^2}{2}}=\frac{2}{t}+1\to \mathrm{\infty }(t\to 0)$$$$\Rightarrow {lim}_{x\to +\mathrm{\infty }}f\left(x\right)=\mathrm{\infty }$$$$$$

Answered by Mathspace last updated on 31/Jan/22

$$2)$$$$letusehospital$$$${lim}_{x\to a}\frac{x^x-a^a}{x-a}$$$$={lim}_{x\to a}{\left(e^{xlnx}\right)}^{\left(1\right)}$$$$={lim}_{x\to a}((lnx+1)e^{xlnx}$$$$=(1+lna)a^a$$

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