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Question Number 108623 by bemath last updated on 18/Aug/20

$$\frac{\supset \mathcal{B}e\mathcal{M}ath\supset }{★}$$$$\left(1\right)\underset{b\to a}{\lim }\frac{b\sqrt{a}-a\sqrt{b}}{a\sqrt{a}+b\sqrt{a}-2a\sqrt{a}}$$$$\left(2\right)\underset{x\to 0}{\lim }\frac{3e^{2x}+e^x-4}{x}$$

Answered by ajfour last updated on 18/Aug/20

$$\lim \frac{b\sqrt{a}-a\sqrt{b}}{a\sqrt{a}+b\sqrt{a}-2a\sqrt{a}}$$$$L=\underset{b\to a}{\lim }\left(\frac{{ab}^2-a^{2}b}{a^3+{ab}^2+2a^{2}b-4a^3}\right)\times \underset{b\to a}{\lim }\left(\frac{a\sqrt{a}+b\sqrt{a}+2\sqrt{a}}{b\sqrt{a}+a\sqrt{b}}\right)$$$$=\underset{b\to a}{\lim }\frac{b(b-a)}{b^2+2ab-3a^2}\times 2$$$$=\underset{b\to a}{\lim }\frac{b(b-a)}{(b+3a)(b-a)}\times 2$$$$\Rightarrow L=\frac{1}{2}$$

Commented by bemath last updated on 18/Aug/20

$$cooll$$

Answered by ajfour last updated on 18/Aug/20

$$L=\underset{x\to 0}{\lim }\frac{3e^{2x}+e^x-4}{x}$$$$L=\underset{x\to 0}{\lim }\frac{3(e^{2x}-1)+(e^x-1)}{x}$$$$L=3\times 2+1=7.$$

Answered by mathmax by abdo last updated on 18/Aug/20

$$2)\mathrm{f}\left(\mathrm{x}\right)=\frac{{3\mathrm{e}}^{2\mathrm{x}}+{\mathrm{e}}^{\mathrm{x}}-4}{\mathrm{x}}\mathrm{we}\mathrm{hsve}{\mathrm{e}}^{2\mathrm{x}}\sim 1+2\mathrm{x}\mathrm{and}{\mathrm{e}}^{\mathrm{x}}\sim 1+\mathrm{x}\Rightarrow$$$$\mathrm{f}\left(\mathrm{x}\right)\sim \frac{3+6\mathrm{x}+1+\mathrm{x}-4}{\mathrm{x}}=7(\mathrm{x}\to 0)\Rightarrow {\lim }_{\mathrm{x}\to 0}\mathrm{f}\left(\mathrm{x}\right)=7$$

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