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COMEDK2024Morning ShiftMathematicsLimitsActual

_ x 0 a+x - a x a(a+x) equals to

Options

  1. A1 2
  2. Ba^ - 3 2
  3. C2 a^ - 3 2
  4. D1 2 a^ 3 2

Correct answer

D. 1 2 a^ 3 2

Step-by-step solution

Let the given limit be L = _ x 0 a+x - a x a(a+x) . Rationalizing the numerator by multiplying and dividing by ( a+x + a ) : L = _ x 0 ( a+x - a )( a+x + a ) x a(a+x) ( a+x + a ) L = _ x 0 (a+x) - a x a(a+x) ( a+x + a ) L = _ x 0 x x a(a+x) ( a+x + a ) L = _ x 0 1 a(a+x) ( a+x + a ) Substituting x = 0 into the expression: L = 1 a(a+0) ( a+0 + a ) L = 1 a^2 ( a + a ) L = 1 a(2 a ) L = 1 2 a a^ 1 2 = 1 2 a^ 3 2 Answer: 1 2 a^ 3 2

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