Limits and continuity

Evaluate the following limit. \[ \lim\limits_{x\to-\infty}\left(-x^4+2x-5\right) \]

Evaluate the following limit. \[ \lim\limits_{x\to 1}\frac{x^2+4x-5}{x^2-4x+3} \]

Given the graph of the function \( f \), find \( \lim\limits_{x\rightarrow 2}f(x) \).

Given the function \(g\) (see the picture), find \(\lim\limits_{x\to \infty}g(x)\). \[ g(x)=\begin{cases} \frac12(x-1)^2+1 & \text{if } x < 1,\\ \frac1{x^2}+2 & \text{if } x \geq 1 \end{cases} \]