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Question Number 93676 by john santu last updated on 14/May/20

$$\mathrm{solve}\mathrm{without}{\mathrm{L}}^{\prime }\mathrm{Hopital}$$$$\underset{x\to 0}{\lim }\frac{5\sqrt{\mathrm{x}+1}-2\sqrt{\mathrm{x}+4}-1}{\mathrm{x}}?$$

Commented by mathmax by abdo last updated on 14/May/20

$$letf\left(x\right)=\frac{5\sqrt{x+1}-2\sqrt{x+4}-1}{x}wehave\sqrt{1+x}\sim 1+\frac{x}{2}$$$$\sqrt{x+4}=2\sqrt{1+\frac{x}{4}}\sim 2(1+\frac{x}{8})\Rightarrow f\left(x\right)\sim \frac{5(1+\frac{x}{2})-2(2+\frac{x}{4})-1}{x}$$$$=\frac{5+\frac{5x}{2}-4-\frac{x}{2}-1}{x}=\frac{2x}{x}=2\Rightarrow {lim}_{x\to 0}f\left(x\right)=2$$

Answered by i jagooll last updated on 14/May/20

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