Getting closer to the powers of \(e\)!

Calculus Level 4

Let \(g:\mathbb{R}\to\mathbb{R}\) be a differentiable function such that \(g(2)=-40\) and \(g^{\prime}(2)=-5\). Then find the value of \(\displaystyle \lim _{ x\to 0 }{ { \left( \dfrac { g\left( 2-{ x }^{ 2 } \right) }{ g\left( 2 \right) } \right) }^{\frac { 4 }{ { x }^{ 2 } } } } \).

Notation:

  • \(g^{\prime}(x)\) denotes the first derivative of \(g(x)\).
  • \(e \approx 2.718\) is the Euler's number.
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