Consider the quadratic function $f\(\left\)(x\(\right\))=-4x^2+32x$. Calculate the slopes of the secant lines between the points $\(\left\)(x,f\(\left\)(x\(\right\))\(\right\))$ and $\(\left\)(4,f\(\left\)(4\(\right\))\(\right\))$, for $x=3.5,3.9,3.99,3.999$.

Consider the position function $s(t) = -12t^2 + 60t$. Approximate the instantaneous velocity at $t=1$ by completing the given table with the average velocities.

A runner's position along a track is recorded at various times during a race. The data is presented in a table. Find the runner's average velocity between the time interval of 0 to 2 seconds.

Identify the vertical asymptote of the function $f\(\left\)(x\(\right\))=\(\frac{x^2-4x+4}{2x-8}\)$.

Determine $\(\displaystyle\) \(\lim\)_{x \(\to\) -2^-}{f(x)}$ using the following graph:

The revenue of a company over time can be modeled by the function $R\(\left\)(t\(\right\))=\(\frac{5000t}{2t+5}\)$. Determine whether a steady-state exists and provide its value as $t$ approaches $\(\infty\)$.

Find the limit of the given function at infinity.

$\(\lim\)_{x\(\to\[\infty\)}\(\left\)(5+\(\frac{101}{x^4}\]\right\))$