What is the limit of the following? \( \lim _{x \rightarrow-\infty} \frac{-2 x^{2}+3 x-4}{3 x^{2}-2 x-1} \)

If \( y=\frac{10-\cos x}{x} \), then simplify completely \( x \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x} \)

\( \frac{d}{dx}((x\times \sin (xy-y^{2})=x^{2}-1)) \)

1. Prove that under the transformation of independent variable \( x= \) \( \varphi(t) \), an \( n^{\text {th }} \)-order linear differential equation remains an \( n^{\text {th }} \)-order linear differential equation, and a homogeneous linear differential equation remains a homogeneous linear differential equation. Here \( x=\varphi(t) \) has continuous derivatives up to order \( n \) and \( \varphi^{\prime}(t) \neq 0 \).

* stirling's Formula \( P_{n}(x)=y_{0}+\frac{\alpha\left(\Delta y_{0}+\Delta y_{-1}\right)}{2}+\frac{\alpha^{2}}{2!} \Delta^{2} y_{-1}+\frac{\alpha\left(\alpha^{2}-i\right)}{3} \) \( \cdot\left(\frac{\Delta^{3} y_{-1}+\Delta^{3} y_{-2}}{2}\right)+\frac{\alpha^{2}\left(\alpha^{2}-i\right)}{4!} \Delta^{4} y_{-2}+\cdots \) \( \alpha=\frac{x-x_{0}}{h}, h=x_{i+1}-x_{i} \) \( P_{n}(x)=\frac{1}{h}\left[\left(\frac{\Delta y_{0}+\Delta y_{-1}}{2}\right)-\frac{1}{6}\left(\frac{\Delta^{3} y_{-1}+\Delta^{3} y_{-2}}{2}\right)\right. \) \( +\frac{1}{30}\left[\frac{\Delta^{5} y_{-2}+\Delta^{5} y_{-3}}{2}\right]+\cdots \)

Integrate \( f(x)=\frac{1}{\left(x^{2}+4\right)\left(x^{2}+2\right)} \) use partial fraction