Evaluate $\lim_{n\to\infty} n^2(1-\cos\frac{x}{n})$ Find this limit by utilizing Lagrange Remainder Theorem:
$$\lim_{n\to\infty} n^2\left(1-\cos\frac{x}{n} \right)$$
I have tried using the definition and can't get it. 

Note that $1-\cos(x/n) = 2 \sin^2\left(\dfrac{x}{2n} \right)$. Hence, we have
$$\lim_{n \to \infty} n^2 \cdot 2 \sin^2\left(\dfrac{x}{2n} \right) = 2\cdot \dfrac{x^2}{2^2} = \dfrac{x^2}2$$
