lim X>0 (1-cos4X+Xsin3X/X*X)

cos(4x) → 1 - 8x² + 32x⁴/3 + ...

sin(3x) → 3x - 9x³/2 + ...

 

so: (1- cos(4x) + x*sin(3x))/x² → (1 - (1 -  8x² + 32x⁴/3 + ...) +x*(3x - 9x³/2 + ...))/x²

 → (8x² + 3x² + higher order terms in x)/x² →  11 + higher order terms in x

 

Hence in the limit as x → 0,  (1- cos(4x) + x*sin(3x))/x² → 11

.

Your question does not make sense.  Perhaps you mean something like this?

 

$$\\\displaystyle\lim_{ X\rightarrow 0} \frac{1-cos4X+Xsin3X}{X*X}\\\\$$