1. Limits and Continuity

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)?

A rock is dropped off the edge of a cliff, and its distance s (in feet) from the top of the cliff after t seconds is s(t)=16t^2. Assume the distance from the top of the cliff to the ground is 96 ft.

a. When will the rock strike the ground? 

Let $g\left(t\right)=\frac{t-9}{\sqrt{t}-3}$.

Make two tables, one showing values of $g$ for $t=8.9,8.99$, and $8.999$ and one showing values of $g$ for $t=9.1,9.01$, and $9.001$.

Let $g\left(t\right)=\frac{t-9}{\sqrt{t}-3}$.

Make a conjecture about the value of ${\displaystyle\lim_{t\to9}\frac{t-9}{\sqrt{t}-3}}$.

Let $g\left(x\right)=\frac{x^3-4x}{8\left|x-2\right|}$. <IMAGE>

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

Let $g\left(x\right)=\frac{x^3-4x}{8\left|x-2\right|}$. <IMAGE>

Make a conjecture about the values of ${\displaystyle\lim_{x\to2^{-}}g\left(x\right)}$, ${\displaystyle\lim_{x\to2^{+}}g\left(x\right)}$, and ${\displaystyle\lim_{x\to2}g\left(x\right)}$ or state that they do not exist.

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.

$f\left(x\right)=\frac{x-2}{\ln \mid x-2\mid }$; $a=2$

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.

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

Determine whether the following statements are true and give an explanation or counterexample.

a. The value of $\underset{x\to 3}{\lim }\frac{x^2-9}{x-3}$ does not exist.

Determine whether the following statements are true and give an explanation or counterexample.

d. $\underset{x\to 0}{\lim }\sqrt{x}$ . (Hint: Graph y=√x)

Determine whether the following statements are true and give an explanation or counterexample.

e. $\underset{x\to \frac{\pi }{2}}{\lim }\cot x=0$ . (Hint: Graph y=cot x)

Sketch the graph of a function with the given properties. You do not need to find a formula for the function. 

f(2) = 1,lim x→2 f(x) = 3

Sketch the graph of a function with the given properties. You do not need to find a formula for the function. 

p(0) = 2,lim x→0 p(x) = 0,lim x→2 p(x) does not exist, p(2)=lim x→2^+ p(x)=1

For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).

a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.