If  f:R\rightarrow R  is a differentiable function and F(2)=6   , then \lim_{x\rightarrow 2}\int_{6}^{f(x)}\frac{2t\:dt}{(x-2)}   is :

 

 

 

  • Option 1)

    24f^{'}(2)

  • Option 2)

    2f^{'}(2)

  • Option 3)

    0

  • Option 4)

    12f^{'}(2)

 

Answers (1)

\\f(2)=6\\\\\:\lim_{x\rightarrow 2}\int_{6}^{f(x)}\frac{2t.dt}{(x-2)}\\\\\:=\lim_{x\rightarrow 2}\frac{t^{2}}{x-2}\; |_{6}^{f(x)}=\lim_{x\rightarrow 2}\frac{\left ( f(x) \right )^{2}-(6)^{2}}{(x-2)}\\\\\:

\\=\lim_{x\rightarrow 2}\frac{2f(x)f^{'}(x)}{1}\\\\\:=2(x).f^{'}(x)\\\\\:2 \times 6\times f^{'}(2)=12f^{'}(2)


Option 1)

24f^{'}(2)

Option 2)

2f^{'}(2)

Option 3)

0

Option 4)

12f^{'}(2)

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