real analysis - Find$limlimits_{nrightarrowinfty}$ $frac{n+sinleft(n^{2}right)}{n+cosleft(nright)}$

Question Find

$$\lim_{n\rightarrow\infty}\frac{n+\sin\left(n^{2}\right)}{n+\cos\left(n\right)}$$

My Approach

$$\lim_{n\rightarrow\infty}\frac{n+\sin\left(n^{2}\right)}{n+cos\left(n\right)}=\lim_{n\rightarrow\infty}\left[\frac{n}{n+\cos\left(n\right)}+\frac{\sin\left(n^{2}\right)}{n+\cos n}\right] =\lim_{n\rightarrow\infty}\left[\frac{1}{1+\frac{\cos\left(n\right)}{n}}+\frac{\sin\left(n^{2}\right)}{n+\cos n}\right]$$

Applying L ' Hospital is not working here

Answer

$$\lim_{n\to\infty}\frac{n+\sin^2(n)}{n+\cos(n)}-1=\lim_{n\to\infty}\frac{\sin^2(n)-\cos(n)}{n+\cos(n)}=0,$$ since the numerator is bounded and the denominator tends to $+\infty$. Therefore $$\lim_{n\to\infty}\frac{n+\sin^2(n)}{n+\cos(n)}=1.$$