The following table gives the position $s\(\left\)(t\(\right\))$ of an object moving along a line at time $t$. Determine the average velocities over the time intervals $\(\left\[\lbrack\)1,1.01\(\right\]\rbrack\)$, $\(\left\[\lbrack\)1,1.001\(\right\]\rbrack\)$, and $\(\left\]\lbrack\)1,1.0001^{}\(\right\).]$. Then make a conjecture about the value of the instantaneous velocity at $t=1$. <IMAGE>

Use a graph of f to estimate $\underset{x\to a}{\lim }f\left(x\right)$ or to show that the limit does not exist. Evaluate f(x) near $x=a$ to support your conjecture.

$f\left(x\right)=\frac{1-\cos (2x-2)}{{(x-1)}^2};a=1$

Consider the position function s(t)=−16t^2+128t (Exercise 13). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=1. <IMAGE>

Determine $\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\))$ and $\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\))$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{40x^5+x^2}{16x^4-2x}$

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

h(x)=e^x(x+1)^3

Evaluate each limit and justify your answer. 

lim x→1 (x+5x / x+2)^4

Evaluate each limit. 

lim x→2 √4x+10 / 2x−2