Chapter 4.6, Problem 23E

Exercises Evaluate the following limits. 1. lim cot x/ln x +01x 2. lim x² In x +014 3. lim x* x0+ 4. lim (cos√√x)1/x +014 5. lim x2/(1-cos x) x10 6. lim e*/* 818 7. lim (secx - tan x) x-x/2- 8. lim [1+(3/x)]* x→∞0

1. Show that f(x) = x3 is not uniformly continuous on R. 2. Show that f(x) = 1/(x-2) is not uniformly continuous on (2,00). 3. Show that f(x)=sin(1/x) is not uniformly continuous on (0,л/2]. 4. Show that f(x) = mx + b is uniformly continuous on R. 5. Show that f(x) = 1/x2 is uniformly continuous on [1, 00), but not on (0, 1]. 6. Show that if f is uniformly continuous on [a, b] and uniformly continuous on D (where D is either [b, c] or [b, 00)), then f is uniformly continuous on [a, b]U D. 7. Show that f(x)=√x is uniformly continuous on [1, 00). Use Exercise 6 to conclude that f is uniformly continuous on [0, ∞). 8. Show that if D is bounded and f is uniformly continuous on D, then fis bounded on D. 9. Let f and g be uniformly continuous on D. Show that f+g is uniformly continuous on D. Show, by example, that fg need not be uniformly con- tinuous on D. 10. Complete the proof of Theorem 4.7. 11. Give an example of a continuous function on Q that cannot be continuously extended to R. 12.…

can I see the steps for how you got the same answers already provided for μ1->μ4. this is  a homework that provide you answers for question after attempting it three tries

1. Prove that for each n in N, 1+2++ n = n(n+1)/2. 2. Prove that for each n in N, 13 +23+ 3. Prove that for each n in N, 1+3+5+1 4. Prove that for each n ≥ 4,2" -1, then (1+x)" ≥1+nx for each n in N. 11. Prove DeMoivre's Theorem: fort a real number, (cost+i sint)" = cos nt + i sinnt for each n in N, where i = √√-1.

Pls help ASAP

2. Sam and Deb have a weekly net income of $1500. They have a pet dog. Their monthly expenses, not related to housing, are $2875. They have savings of $32 000. They are considering two housing options: Option 1: Renting a 2-bedroom condo for $1650 a month, plus utilities averaging $210 a month Option 2: Buying a 2-bedroom condo for a down payment of $24 500, bi-weekly mortgage payments of $1100, and a monthly condo fee of $475 a) Determine the monthly cost of each housing option. Factoring in other expenses not related to housing, which one can Sam and Deb afford? b) Suppose their dog falls ill and they have to pay $85 every week to cover veterinarian and medical expenses. Calculate the additional monthly expenses. How much money would be available for savings if they choose housing option 2?

I bought sparrows at 3 for a penny, turtle doves at 2 for a penny, anddoves at 2 pence each. If I spent 30 pence buying 30 birds and boughtat least one of each kind of bird, how many birds of each kind did I buy?(This is a problem from Fibonacci’s Liber Abaci, 1202.)