1. Limits and Continuity

Given the graph of a function $f$, find a number $\epsilon >0$ such that if $|x-1|<\epsilon$, then $\left|f\right(x)-1|<0.2$.

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

Make a conjecture about the value of ${\(\displaystyle\]\lim\)_{t\(\to\)9}\(\frac{t-9}{\sqrt{t}\)-3}}$.

For the function $f\left(x\right)=2x^2+3x$, what is the average rate of change from $x=0$ to $x=5$?

The hyperbolic cosine function, denoted $\(\cosh\]\left\)(x\(\right\))$, is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $\(\cosh\]\left\)(x\(\right\))=\(\frac{e^{x}\)+e^{-x}}{2}$.

b. Evaluate $\cosh \left(0\right)$. Use symmetry and part (a) to sketch a plausible graph for $y=\cosh \left(x\right)$.

Using the Sandwich Theorem

a. It can be shown that the inequalities 1 − x²/ 6 < (x sin x) / (2−2cos x) < 1 hold for all values of x close to zero (except for x = 0). What, if anything, does this tell you about limx→0 (x sin x) / (2 − 2cos x)?

Give reasons for your answer.

[Technology Exercise] b. Graph y = 1 − (x²/6), y=(x sinx)/(2 − 2cos x), and y = 1 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.

Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 2x / (x² − 1)

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. $[0, 1]$

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x sin (1/x) = 0

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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→−3 |2x|=6 (Hint: Use the inequality ∥a|−|b∥≤|a−b|, which holds for all constants a and b (see Exercise 74).)

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

c. limx→1 f(x) does not exist.

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x² sin (1/x) = 0

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Complete the following steps for the given functions. 

b. Find the vertical asymptotes of $f$ (if any).

$f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)$

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→4 x^2−16 / x−4=8 (Hint: Factor and simplify.)

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>

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

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