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Let f(x) be a twice-differentiable funct...

Let `f(x)` be a twice-differentiable function and `f''(0)=2.` Then evaluate `lim_(xto0) (2f(x)-3f(2x)+f(4x))/(x^(2)).`

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To evaluate the limit \[ \lim_{x \to 0} \frac{2f(x) - 3f(2x) + f(4x)}{x^2}, \] given that \( f''(0) = 2 \), we will use Taylor series expansion around \( x = 0 \). ...
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