Limits

Given $f(x) = f(-x)$ That means, $f(x)$ is an even function. $f(x) = a_{2n}x^{2n} +a_{2n-2}x^{2n-2}+\cdots + a_4x^4 + a_2x^2\\f'(x) = \cdots + 2a_2x\\f'''(x)=\cdots + 24a_4x\\\Rightarrow f'(0) = 0\\f'''(0) = 0$

Also, $\lim_{x\to0}\dfrac{\sin x}{\sqrt[3] x} = \lim_{x\to0}\dfrac{\sqrt[3] {x^2}\sin x}{ x} = 0\times 1 = 0$ $\lim_{x\to0}e^x = 1$

Therefore, $\lim_{x\to0} \dfrac{ f'(x) }{ f'''(x) + \frac{\sin x}{\sqrt[3]{x}} + e^x} = \dfrac0{0+0+1} = 0$