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Question Number 184823 by mnjuly1970 last updated on 12/Jan/23

$$$$$${\mathrm{Lim}}_{x\to 0^{+}}\frac{1-\cos (1-\cos \left(\sqrt{x}\right))}{x^4}$$

Answered by cortano1 last updated on 12/Jan/23

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

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