1. Limits and Continuity

a. Estimate lim x→π/4 cos 2x / cos x − sin x by making a table of values of cos 2x / cos x − sin x for values of x approaching π/4. Round your estimate to four digits.

Evaluate lim x→∞ f(x) and lim x→−∞ f(x) sing the figure. <IMAGE>

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {x^2+1 if x≤−1

3 if x>−1; a=−1

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {√x if x<4

3 if x=4; a=4

x+1 if x>4

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2−25 / x−5; a=5

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2+x−2 / x−1; a=1

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)

Postage rates Assume postage for sending a first-class letter in the United States is $0.47 for the first ounce (up to and including 1 oz) plus $0.21 for each additional ounce (up to and including each additional ounce).

a. Graph the function p=f(w) that gives the postage p for sending a letter that weighs w ounces, for 0<w≤3.5.

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^− tan x

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^+ tan x

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^− tan x

Find polynomials p and q such that f=p/q is undefined at 1 and 2, but f has a vertical asymptote only at 2. Sketch a graph of your function.

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

The line x=−1 is a vertical asymptote of the function f(x) =x^2 − 7x + 6 / x^2 − 1.

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

f(x)=x^2−3x+2 / x^10−x^9

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

h(x)=e^x(x+1)^3