1. Limits and Continuity

Suppose f(x) lies in the interval (2, 6). What is the smallest value of ε such that |f (x)−4|<ε?

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.

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

a. The graph of a function can never cross one of its horizontal asymptotes.

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 4 / x²/⁵

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

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→∞ (2√x + x⁻¹) / (3x − 7)

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

h. f(0)=0

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

$h\(\left\)(4\(\right\))$

Using the graph, find the specified limit or state that the limit does not exist (DNE).

$\(\lim\)_{x\(\rightarrow\)0^{-}}f\(\left\)(x\(\right\))$ , $\(\lim\)_{x\(\rightarrow\)0^{+}}f\(\left\)(x\(\right\))$, $\(\lim\)_{x\(\rightarrow\)0}f\(\left\)(x\(\right\))$

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

f(x)=1/ √x sec x

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→−1 f(x)

Finding Limits Graphically

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

c. limx→0− f(x) = 0

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

Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.

a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?

Average Rates of Change

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

g(t)=2+cos t

b. [0,π]