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Question Number 133474 by Abdoulaye last updated on 22/Feb/21

Commented by Abdoulaye last updated on 22/Feb/21

$$please$$

Answered by Olaf last updated on 22/Feb/21

$$e^{\sqrt{x+1}}-e^{\sqrt{x}}=e^{\frac{\sqrt{x+1}+\sqrt{x}}{2}}[e^{\frac{\sqrt{x+1}-\sqrt{x}}{2}}-e^{-\frac{\sqrt{x+1}-\sqrt{x}}{2}}]$$$$=2e^{\frac{\sqrt{x+1}+\sqrt{x}}{2}}\sinh \left(\frac{\sqrt{x+1}-\sqrt{x}}{2}\right)$$$$=2e^{\frac{\sqrt{x+1}+\sqrt{x}}{2}}\sinh \left(\frac{1}{2(\sqrt{x+1}+\sqrt{x})}\right)$$$$\underset{\mathrm{\infty }}{\sim }2e^{\sqrt{x}}\sinh \left(\frac{1}{4\sqrt{x}}\right)\underset{\mathrm{\infty }}{\sim }2e^{\sqrt{x}}\times \frac{1}{4\sqrt{x}}=\mathrm{\infty }$$

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