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Differential equation hidden in a limit (definition of the derivative)

This type of question, asked 3 times: 29 Jan 2024, 5 Apr 2024, 24 Jan 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    29 Jan 2024

    177
    Let f(x)=lim⁡r→x{2r2[(f(r))2−f(x)f(r)]r2−x2−r3ef(r)r}f(x) =\sqrt{\lim_{r\rightarrow x} \left\{ \frac{2r^{2}\left\lbrack (f(r))^{2}-f(x)f(r) \right\rbrack}{r^{2}-x^{2}}-r^{3}e^{\frac{f(r)}{r}} \right\}} be differentiable in (−∞,0)∪(0,∞)( - \infty,0) \cup (0,\infty) and f(1)=1f(1) = 1. Then the value of ea, such that f(a)=0f(a) = 0, is equal to

  2. Question 2

    5 Apr 2024

    86
    Let ff be a differentiable function in the interval (0,∞)(0,\infty) such that f(1)=1f(1) = 1 and lim⁡t→xt2f(x)−x2f(t)t−x=1\lim_{t\rightarrow x} \frac{t^{2}f(x) -x^{2}f(t)}{t-x}= 1 for each x>0x> 0. Then 2f(2)+3f(3)2f(2) + 3f(3) is equal to

  3. Question 3

    24 Jan 2026 S2

    66
    Let y=y(x)y = y(x) be a differentiable function in the interval (0,∞)(0,\infty) such that y(1)=2y(1) = 2. and lim⁡t→x(t2y(x)−x2y(t)x−t)=3\lim_{t \rightarrow x} \left( \frac{t^{2}y(x) -x^{2}y(t)}{x - t} \right)= 3 for each x>0x > 0. Then 2y(2)2y(2) is equal to

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