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?
limX→01−cos4X+Xsin3XX∗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
.