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Question Number 56453 by wasim last updated on 16/Mar/19

Commented by wasim last updated on 16/Mar/19

$$\mathrm{please}\mathrm{solve}\mathrm{question}7$$$$$$$$\mathrm{thanks}$$

Commented by maxmathsup by imad last updated on 16/Mar/19

$$wehavef\left(x\right)=\frac{{cos}^{2}x-{sin}^{2}x-1}{\sqrt{x^2+1}-1}ifx\ne 0andf\left(0\right)=a$$$$fcontinueat0\iff {lim}_{x\to 0}f\left(x\right)=f\left(0\right)=abutwehave$$$$f\left(x\right)=\frac{cos\left(2x\right)-1}{\sqrt{1+x^2}-1}andcos\left(2x\right)\sim 1-\frac{4x^2}{2}=1-2x^2(x\to 0)\Rightarrow cos\left(2x\right)-1\sim -2x^2$$$$\sqrt{1+x^2}\sim 1+\frac{1}{2}{x}^2\Rightarrow \sqrt{1+x^2}-1\sim \frac{x^2}{2}\Rightarrow f\left(x\right)\sim -4(x\to 0)\Rightarrow$$$${lim}_{x\to 0}f\left(x\right)=-4=a.$$

Answered by ajfour last updated on 16/Mar/19

$$\underset{x\to 0}{\lim }[f\left(x\right)=\frac{1-\cos 2x}{1-\sqrt{x^2+1}}]=a$$$$=\underset{x\to 0}{\lim }\frac{2\sin {}^{2}x}{1-(1+\frac{x^2}{2})}=a$$$$=-4\underset{x\to 0}{\lim }{\left(\frac{\sin x}{x}\right)}^2=a$$$$\Rightarrow a=-4\times 1=-4.$$

Commented by wasim last updated on 17/Mar/19

$$\mathrm{thanks}$$

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