5 Lim n->inf (n²*c³)/(1+2n²*x²)

The given expression is $\frac{n^2 x^3}{1 + 2n^2 x^2}$ and we need to find the limit as $n \to \infty$.

$$\lim_{n \to \infty} \frac{n^2 x^3}{1 + 2n^2 x^2}$$

As $n \to \infty$, the term $\frac{1}{n^2}$ approaches 0.

$$\frac{x^3}{0 + 2x^2}$$ $$\frac{x^3}{2x^2}$$

Assuming $x \neq 0$:

$$\frac{x^3}{2x^2} = \frac{x}{2}$$

If $x=0$, the original expression is $\lim_{n \to \infty} \frac{n^2 \cdot 0^3}{1 + 2n^2 \cdot 0^2} = \lim_{n \to \infty} \frac{0}{1} = 0$. Our result $\frac{x}{2}$ also gives 0 for $x=0$.