lim-x-pi-2-2tan-2-x-3-2tan-2-x-10-cot-x-

Question Number 163966 by bobhans last updated on 12/Jan/22

\lim_{x \to \frac{π}{2}} \frac{\sqrt{2 \tan^{2} x + 3} - \sqrt{2 \tan^{2} x + 10}}{\cot x} = ?

Answered by cortano1 last updated on 12/Jan/22

s e t x = \frac{π}{2} + h \Rightarrow {\begin{cases} \tan x = - \cot h \\ \cot x = - \tan h \end{cases}

\lim_{h \to 0} \frac{\sqrt{2 \cot^{2} h + 3} - \sqrt{2 \cot^{2} h + 10}}{- \tan h}

= \lim_{h \to 0} \frac{\sqrt{2 + 3 \tan^{2} h} - \sqrt{2 + 10 \tan^{2} h}}{- \tan^{2} h}

= \sqrt{2} \times \lim_{h \to 0} \frac{\sqrt{1 + \frac{3 \tan^{2} h}{2}} - \sqrt{1 + 5 \tan^{2} h}}{- \tan^{2} h}

= - \sqrt{2} \times \lim_{h \to 0} \frac{(\frac{3}{4} - \frac{5}{2}) \tan^{2} h}{\tan^{2} h}

= - \sqrt{2} \times (- \frac{7}{4}) = \frac{7 \sqrt{2}}{4}

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