1. Limits and Continuity

Given the curve $r\left(t\right)=(3t,4t)$, for $t\ge 0$, starting from the point $t=0$, which of the following is the correct reparametrization of the curve in terms of arclength $s$?

If a function f represents a system that varies in time, the existence of lim $\underset{t\to \infty }{\lim }f\left(t\right)$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a bacteria culture is given by $p\left(t\right)=\frac{2500}{t+1}$.

For which values of $p$ does the series $\sum _{n=1}^{\infty }\frac{{(-1)}^{n+1}}{n^p}$ converge?

Finding Limits Graphically

Which of the following statements about the function y = f(x) graphed here are true, and which are false?

j. limx→2− f(x) = 2

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→1 x^4=1

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = 3x - 4, x = 2

Average Rates of Change

In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.

P(θ)=θ³ − 4θ² + 5θ; [1,2]

Theory and Examples

Once you know limx→a+ f(x) and limx→a− f(x) at an interior point of the domain of f, do you then know limx→a f(x)? Give reasons for your answer.

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.

For the function $f\left(x\right)=2x$, what is the limit of the average rate of change of $f\left(x\right)$ as $x$ increases from $2$ to $4$?

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.

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

Theory and Examples

Suppose that g(x) ≤ f(x) ≤ h(x) for all x≠2 and suppose that lim x→2 g(x) = lim x→2 h(x) = −5. Can we conclude anything about the values of f, g, and h at x = 2? Could f(2) = 0? Could limx→2 f(x)=0? Give reasons for your answers.

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>

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let f(x)=(x² − 1)/(|x| − 1).

b. Support your conclusion in part (a) by graphing f near c = -1 and using Zoom and Trace to estimate y-values on the graph as x→−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>

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