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Limits with sqrt(2) - sqrt(1 + cos x) in the denominator

This type of question, asked 2 times: 27 Jan 2024, 4 Apr 2024. All 2 questions below, oldest first.

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  1. Question 1

    27 Jan 2024

    78
    If a=lim⁡x→01+1+x4−2x4a =\lim_{x \rightarrow 0} \frac{\sqrt{1 +\sqrt{1 +x^{4}}}-\sqrt{2}}{x^{4}} and b=lim⁡x→0sin⁡2x2−1+cos⁡xb =\lim_{x \rightarrow 0} \frac{\sin^{2}x}{\sqrt{2}-\sqrt{1 + \cos x}}, then the value of ab3ab^{3} is :

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  2. Question 2

    4 Apr 2024

    151
    If the function f(x)={72x−9x−8x+12−1+cos⁡x,x≠0alog⁡e2log⁡e3,x=0f(x) =\left\{ \begin{matrix} \frac{72^{x}-9^{x}-8^{x}+ 1}{\sqrt{2}-\sqrt{1 + \cos x}} & ,x \neq 0 \\ a\log_{e}2\log_{e}3 & ,x = 0 \end{matrix} \right. is continuous at x=0x = 0, then the value of a2a^{2} is equal to

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