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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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The given limit has 0/0 form.
Using L'Hospital's rule, we have
Limit= `underset(xto0)lim(2f'(x)-6f'(2x)+4f'(4x))/(2x)" "`(0/0 form)
`=underset(xto0)lim(2f''(x)-12f''(2x)+16f''(4x))/(2)`
(Using L'Hospital's rule)
`=(6f''(0))/(2)=6`
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