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Manipal MET2012MathematicsLimits

Let f: R R be such that f(1)=3 and f^ (1)=6 . Then, _ x 0 [ f(1+x) f(1) ]^ 1 / x equals

Options

  1. A1
  2. Be^ 1 / 2
  3. Ce^2
  4. De^3

Correct answer

C. e^2

Step-by-step solution

aligned & Let y= [ f(1+x) f(1) ]^ 1 / x & y= 1 x [ f(1+x)- f(1)] & _ x 0 y= _ x 0 [ f(1+x)- f(1)] x & _ x 0 y= _ x 0 [ 1 f(1+x) f^ (1+x) ] & (using L' Hospital's rule) & = f^ (1) f(1) = 6 3 =2 & _ x 0 y=e^2 aligned

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