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Question Number 168911 by bagjagugum123 last updated on 21/Apr/22

Commented by infinityaction last updated on 21/Apr/22

$$\underset{x\to a}{\lim }\frac{(x^2-a^2)\sin (x-a){(\sqrt{x+b}+\sqrt{a+b})}^2}{{(\sqrt{x+b}-\sqrt{a+b})}^2{(\sqrt{x+b}+\sqrt{a+b})}^2}$$$$\underset{x\to a}{\lim }\frac{(x^2-a^2)\sin (x-a){(\sqrt{x+b}+\sqrt{a+b})}^2}{{(x-a)}^2}$$$$\underset{x\to a}{\lim }\frac{(x+a)\sin (x-a){(\sqrt{x+b}+\sqrt{a+b})}^2}{(x-a)}$$$$2a{\left(2\sqrt{a+b}\right)}^2$$$$8a(a+b)$$$$$$$$$$

Answered by qaz last updated on 21/Apr/22

$$\underset{\mathrm{x}\to \mathrm{a}}{\lim }\frac{({\mathrm{x}}^2-{\mathrm{a}}^2)\sin (\mathrm{x}-\mathrm{a})}{{(\sqrt{\mathrm{x}+\mathrm{b}}-\sqrt{\mathrm{a}+\mathrm{b}})}^2}=\underset{\mathrm{x}\to 0}{\lim }\frac{({\mathrm{x}}^2+2\mathrm{ax})\sin \mathrm{x}}{{(\sqrt{\mathrm{x}+\mathrm{a}+\mathrm{b}}-\sqrt{\mathrm{a}+\mathrm{b}})}^2}$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{({\mathrm{x}}^2+2\mathrm{ax})\sin \mathrm{x}}{(\mathrm{a}+\mathrm{b}){(\sqrt{1+\frac{\mathrm{x}}{\mathrm{a}+\mathrm{b}}}-1)}^2}=\underset{\mathrm{x}\to 0}{\lim }\frac{{2\mathrm{ax}}^2+\mathrm{o}\left({\mathrm{x}}^2\right)}{(\mathrm{a}+\mathrm{b})[\frac{{\mathrm{x}}^2}{4{(\mathrm{a}+\mathrm{b})}^2}+\mathrm{o}\left({\mathrm{x}}^2\right)]}$$$$=8\mathrm{a}(\mathrm{a}+\mathrm{b})$$

Commented by bagjagugum123 last updated on 21/Apr/22

$$thankyouSir$$

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