1. Limits and Continuity

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

${\lim }_{x\to 4}g\left(x\right)$

Use the graph of $g$ in the figure to find the following values or state that they do not exist. <IMAGE>

${\displaystyle\lim_{x\to0}g\left(x\right)}$

Use the graph of $f$ in the figure to find the following values or state that they do not exist. <IMAGE>

$f\left(1\right)$

Use the graph of $f$ in the figure to find the following values or state that they do not exist. <IMAGE>

$f\left(0\right)$

Let $f\left(x\right)=\frac{x^2-4}{x-2}$ . <IMAGE>

Calculate $f\left(x\right)$ for each value of $x$ in the following table.

Let $f\left(x\right)=\frac{x^2-4}{x-2}$. <IMAGE>

Make a conjecture about the value of ${\displaystyle\lim_{x\to2}\frac{x^2-4}{x-2}}$.

The function $s(t)$ represents the position of an object at time t moving along a line. Suppose $s(2)=136$ and $s(3)=156$ . Find the average velocity of the object over the interval of time $[2, 3]$ .

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

a. $[0, 2]$

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

c. $[0, 1]$

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time. 

a. s(t)=−16t^2+80t+60 at t=3

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time. 

c. s(t)=40 sin 2t at t=0

Tangent lines with zero slope

a. Graph the function f(x)=x^2−4x+3.

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

a. Graph the position function, for 0≤t≤9.

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time t=a.

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

d. For what values of t on the interval [0, 9] is the instantaneous velocity positive (the projectile moves upward)?