1. Limits and Continuity

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→7 f(x)=9, where f(x)={3x−12 if x≤7

x+2 if x>7

Domains and Asymptotes

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

y = x³ / (x³ − 8)

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\[\lbrack\)0,h\(\right\]\rbrack\)$, where $h\(\gt{0}\)$ is a real number

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = 2x/(x + 1)

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

lim x→−∞ cot^−1x

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = (x + 3)/(x + 2)

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

$f\left(x\right)=1-\ln x$

Let f(x) =x^2−2x+3.

a. For ε=0.25, find the largest value of δ>0 satisfying the statement

|f(x)−2|<ε whenever 0<|x−1|<δ.

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.

c. How does the graph behave near x = 1 and x = −1?

Give reasons for your answers.

y = (3/2)(x − (1 / x))²/³

Use the precise definition of infinite limits to prove the following limits.

$\underset{x\to 0}{\lim }(\frac{1}{x^4}-\sin \left(x\right))=\infty$

Finding Limits Graphically

Let f(x) = {3 - x , x < 2

2, x = 2

x/2, x > 2

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a. Find limx→2+ f(x), limx→2− f(x), and f(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\[\lbrack\)1,2\(\right\]\rbrack\)$

Determine the following limits at infinity.

lim t→∞ (−12t^−5)

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 tan θ / θ²cot 3θ

A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.

lim x→−2^− f(x)