Multiple choice

If $f(x)=\displaystyle \frac { 4-7x }{ 7x+4 } $, $\underset{ x\rightarrow 0 }{Lim}{ f(x)=l } $ and $\underset{ x\rightarrow \infty }{Lim}{ f(x)=m } $ the quadratic equation having roots as $\displaystyle \dfrac{1}{l}$ and $\displaystyle \dfrac{1}{m}$ is

  1. $ x^{2}-1=0 $
  2. $ x^{2}-x+1=0 $
  3. $x^2-\dfrac{1}{2}=0$
  4. $ x^{3}-1=0 $
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A Correct answer
Explanation

l = lim(x->0) (4-7x)/(7x+4) = 4/4 = 1. m = lim(x->inf) (4-7x)/(7x+4) = -7/7 = -1. Roots are 1/l = 1 and 1/m = -1. Equation: (x-1)(x+1) = x^2 - 1 = 0.

AI explanation

First, find the limit as x approaches 0, which is l = (4 - 0) / (0 + 4) = 1. Next, find the limit as x approaches infinity by dividing the numerator and denominator by x, giving m = lim (-7) / 7 = -1. The required roots for the new quadratic equation are 1/l = 1 and 1/m = -1. The quadratic equation with roots 1 and -1 is x^2 - 1 = 0.

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