1. Limits and Continuity

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→3 f(x)

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 $\left\lbrack1,1.01\right\rbrack$, $\left\lbrack1,1.001\right\rbrack$, and $\left\lbrack1,1.0001^{}\right.]$. Then make a conjecture about the value of the instantaneous velocity at $t=1$. <IMAGE>

Given the function $f\left(x\right)=-16x^2+64x$, complete the following. <IMAGE>

Find the slopes of the secant lines that pass though the points $\left(x,f\left(x\right)\right)$ and $\left(2,f\left(2\right)\right)$, for $x=1.5,1.9,1.99,1.999,$ and $1.9999$ (see figure).

Given the function $f\left(x\right)=-16x^2+64x$, complete the following. <IMAGE>

Make a conjecture about the value of the limit of the slopes of the secant lines that pass through $\left(x,f\left(x\right)\right)$ and $\left(2,f\left(2\right)\right)$ as $x$ approaches $2$.

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-16t^2+128t$. Find the average velocity of the object over the following intervals.

$\left\lbrack1,4\right\rbrack$

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-16t^2+128t$. Find the average velocity of the object over the following intervals.

$\left\lbrack1,2\right\rbrack$

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-4.9t^2+30t+20$. Find the average velocity of the object over the following intervals.

$\left\lbrack0,3\right\rbrack$

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-4.9t^2+30t+20$. Find the average velocity of the object over the following intervals.

$\left\lbrack0,h\right\rbrack$, where $h\gt{0}$ is a real number

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

${\lim }_{x\to 1^{-}}f\left(x\right)$

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

${\lim }_{x\to 1^{+}}f\left(x\right)$

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

a. f(1)

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

d. ${\lim }_{x\to 1}f\left(x\right)$

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

h. ${\lim }_{x\to 3}f\left(x\right)$

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

l. ${\lim }_{x\to 2}f\left(x\right)$

Consider the position function s(t) =−16t^2+100t representing the position of an object moving vertically along a line. Sketch a graph of s with the secant line passing through (0.5, s(0.5)) and (2, s(2)). Determine the slope of the secant line and explain its relationship to the moving object.