Calculus: Early Transcendentals, Chapter 4, 4.4, Section 4.4, Problem 49

lim_(x->1) (x/(x-1) - 1/(ln(x)))
To solve, plug-in x=1.
lim_(x->1) (x/(x-1) - 1/(ln(x))) = 1/(1-1)-1/ln(1)=1/0-1/0=oo-oo
Since the result is indeterminate, we can apply the L'Hospital's Rule. To do so, subtract the two rational expressions.
lim_(x->1) (x/(x-1)-1/(ln(x))) = lim_(x->1) ((xln(x))/((x-1)ln(x)) - (x-1)/((x-1)ln(x))) = lim_(x->1) (xln(x) - x + 1)/((x-1)ln(x))
Now that we have one rational function, take the derivative of the numerator and denominator.
lim_(x->1) ((xln(x) - x + 1)')/(((x-1)ln(x))')= lim_(x->1) (x*1/x + 1*ln(x) - 1 + 0)/((x - 1)*1/x + 1*ln(x))= lim_(x->1) (1+lnx-1) /(1-1/x+ln(x))=lim_(x->1) ln(x)/(1-1/x+ln(x))
Then, plug-in x = 1.
= ln(1)/(1-1/1+ln1) = 0/0
Since the result is still indeterminate, apply the L'Hospital's Rule again. So, take the derivative of the numerator and denominator.
lim_(x->1) ((ln(x))')/((1-1/x+ln(x))') = lim_(x->1) (1/x)/(0+1/x^2+1/x)= lim_(x->1) (1/x)/(1/x^2+1/x)= lim_(x->1) x/(1+x)
Then, plug-in x=1.
=1/(1+1) = 1/2

Therefore, lim_(x->1) (x/(x-1)-1/ln(x)) = 1/2 .

How did the Happy Prince compensate for his ignorance of his people's suffering?

To compensate for his lifelong ignorance of his people's suffering, the "Happy Prince" did what he still could to recognize and lessen their misery. He started by asking the Swallow to take the ruby from his sword-hilt to a seamstress who needed oranges for her sick son. He then gave up each of his sapphire eyes to help two people obtain relief from the cold.
After sacrificing his eyesight, the Prince did not let himself become blind to poverty and pain again. Instead he asked the Swallow to fly all over the city, come back, and tell him about the suffering he had seen.
When the Swallow returned from his trip, he told the Prince about beggars sitting outside rich people's gates and cold, starving children in dark lanes. In response, the Prince instructed him to pick all the gold off his statue and take it to the poor so they could eat and be happy for a while.
The Happy Prince made up for his former blindness to want and sorrow by giving up his splendor, and even his sight, to brighten the lives of the poorest people in his city.

Discuss the poet's idea of love as revealed in the poem "Sweetest Love, I Do Not Goe" by John Donne.

The poem by John Donne (1572–1631) that opens with the line "Sweetest love, I do not go" is entitled "Song." In order to understand the poet's idea of love as expressed in this work, it is important to know the meaning of certain words and phrases in the context of what they meant in Donne's time.
In the first stanza, it helps to know that the word "use" here means to become accustomed to, or "used to." "In jest" is another way of saying "by imitation." The first stanza describes a parting of the poet from his beloved. He does not go because he has grown tired of her or in hopes of finding someone "fitter" (better). This parting might as well be a way for the poet to get used to the final separation of death (a subject that comes up often in the thoughts of Donne and other writers of his time).
In the second stanza, Donne uses the symbol of the constancy (reliability) of the sun's daily journey across the heavens and reappearance after night time to explain that he, too, will hurry back in a way that the sun cannot, since his emotions (something the sun lacks) will drive him on. The "wings and spurs" the poet "takes" are metaphors for his passion or devotion.
In the third stanza, Donne laments the human limitation of powerlessness over time. In the fourth stanza, he tells his beloved that when she sighs (each sigh was then believed to use up a drop of blood) and weeps, he is hurt because she feels pain.
So far, we see that the poet's idea of love is that in loving, he experiences a parting that is practice for death and that when his loved one hurts, he feels what she is feeling. Donne describes the latter situation with the oxymoron "unkindly kind." In other words, her distress causes him to experience distress. We see that great empathy is part of Donne's idea of love.
In the fifth stanza, "divining" refers to the ancient practice of foreseeing the future through certain rituals; this implies a bit of superstition. The poet asks the beloved not to worry that bad things may happen ("Forethink me any ill"), for "Destiny may take thy part" and fulfill her fears. In other words, she should not tempt fate by worrying that something bad may happen to him. Instead, the beloved should think of him while parted as she would if they were asleep side by side. "They who one another keep / Alive, ne'er parted be" is a powerful concluding line. Nothing can ever truly part a pair whose love for each other is so strong it feels like it is their life's force.

To determine Donne’s idea of love in the poem, consider first how he explains to his lover his reason for leaving. It is not “For weariness of thee” or “in hope the world can show, A fitter love for me.” Instead, he notes that he must die and that he prefers “to use myself in jest.” Does this perhaps suggest that he feels love carries with it certain obligations or duties that go beyond his personal feelings for someone? Donne is reassuring in the second stanza, contrasting his love with the actions of the sun. He says that the sun set the previous day, but he has returned once again today. The sun has returned, even though it has “no desire nor sense,” in contrast to Donne, who has love. The point he is making should be clear if you bear this in mind when reading the second half of the third stanza, where he urges his lover to believe that he will make “speedier journeys” than the sun. Take a look at the fourth stanza and you will see how Donne discusses the brief, or even fleeting, character of human life and how men’s powers are weak. If he stays with his lover, his life blood will “decay” and she will sigh “my soul away.” This does not suggest a retention of love but rather a wasting away of human life. On the other hand, Donne goes on to lay out how he thinks love will be preserved if they part. In the final stanza, he concludes with an attempt to console, writing, “But think that we, Are but turned aside to sleep.” Here Donne is trying to convey again the sense of the parting being temporary, while love remains eternal. And he finishes this idea in the final two lines: “They who one another keep, Alive, ne’er parted be."
https://www.poetryfoundation.org/poems/44128/song-sweetest-love-i-do-not-go

What does the police officer do after saying good night to the man in the doorway?

After the police officer has said good night to Bob, the man waiting outside the hardware store, he continues pounding the beat. All very normal, we might think. Certainly, that's how 'Silky' Bob feels. But that's not all the police officer did. Judging by what happens right at the end of the story, the officer must also have written a note to Bob and handed it to a plain-clothes detective. The police officer is none other than Jimmy Wells, Bob's old friend from all those years ago, and the man for whom he was waiting outside the hardware store.
As an officer of the law, Jimmy does his duty. But he must still retain some feelings of friendship towards Bob. That's why he can't arrest Bob himself and has to get a plain-clothes detective to do it instead.

dy/dx = y + 3 Solve the differential equation

Recall that in solving simple first order "ordinary differential equation" (ODE),  we may apply variable separable differential equation wherein:
N(y)y'=M(x)
N(y)(dy)/(dx)=M(x)
N(y) dy=M(x) dx
Before we can work on the direct integration: int N(y) dy= int M(x) dx to solve for the  general solution of a differential equation.
 For the given first order ODE: (dy)/(dx)=y+3 can be rearrange by cross-multiplication into:
(dy)/(y+3)=dx
Apply direct integration on both sides: int(dy)/(y+3)=int dx
 For the left side, we consider u-substitution by letting:
u= y+3 then du = dy
 
The integral becomes:
 int(dy)/(y+3)=int(du)/(u)
 Applying basic integration formula for logarithm:
 int(du)/(u)= ln|u|
 Plug-in u = y+3 on ln|u | , we get:
 int(dy)/(y+3)=ln|y+3|
For the right side, we apply the basic integration: int dx= x+C
  
Combing the results from both sides, we get the general solution of the differential equation as:
ln|y+3|= x+C
or
y =e^(x+C)-3
y =Ce^x-3

Please list examples of when Holden feels isolated or depressed and alienated from his peers.

1. In chapter 6, Holden has a physical altercation with Stradlater and gets punched in the mouth. In the next chapter, Holden decides to hang out in Ackley's room to get away from Stradlater, and Ackley gives him a hard time for laying in his roommate's bed. When Ackley begins to question Holden about the fight, Holden gets up and stares out the window. Holden then mentions, "I felt so lonesome, all of a sudden. I almost wished I was dead" (Salinger, 26). Holden leaves shortly after arguing with Ackley and heads to New York City, where he continues to feel isolated and depressed.
2. In chapter 14, Holden has an extremely awkward interaction with a young prostitute named Sunny. After Holden pays Sunny, she returns with her pimp, Maurice, who demands another five dollars. Holden initially challenges Maurice and ends up getting punched in the stomach. After Maurice and Sunny leave, Holden once again feels isolated and depressed. Holden mentions, "What I really felt like, though, was committing suicide. I felt like jumping out the window" (Salinger, 57).
3. Following Holden's uncomfortable interaction with Mr. Antolini, he ends up sleeping in the waiting room of Grand Central Station. The next day, Holden walks down Fifth Avenue and begins feeling extremely anxious and alienated. Each time Holden is about to step off a curb, he begins feeling like he will disappear. Holden then starts speaking to his dead brother by saying, "Allie, don't let me disappear. Allie, don't let me disappear. Allie, don't let me disappear. Please, Allie" (Salinger, 106).

Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 12

The graph of the position function of two runners $A$ and $B$ is shown below. Suppose that the runners finish a 100$m$ race in a tie.



a.) Describe and compare how the runners run the race.
Based from the graph, the runner $A$ finishes the race with a constant speed while runner $B$ has changing speed.

b.) When is the time where the distance between the runners is greatest?
Based from the graph, the time where the distance between the runners is greatest, is approximately at $t = 5$ seconds.

c.) At what time do the runners have the same velocity?
Based from the graph, the time where the runners have the same velocity is at $t = 0 $ and $t = 14 $ seconds.