Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 21

Explain using theorems of continuity why the function $\displaystyle F(x) = \frac{x}{x^2 + 5x + 6}$ is continuous at every number in its domain. State the domain.

The function $F(x)$ is a rational function that is continuous on its domain according to the definition. The function is defined for all values of $x$ except for the value that will make denominator equal to 0. So,

$x^2 + 5x + 6 = 0$

Using Quadratic Formula,


$
\begin{equation}
\begin{aligned}
x_{(1,2)} & = \frac{-b \pm \sqrt{b^2 - 4 ac }}{2a}\\
x_{(1,2)} & = \frac{-5 \pm \sqrt{(5)^2 - 4(1)(6)}}{2(1)}
\end{aligned}
\end{equation}
$


$x = -2 \quad \text{ and } \quad x = -3$

Therefore,
Domain: $(-\infty, -3) \bigcup (-3, -2) \bigcup (-2, \infty)$

Comments

Post a Comment

Popular posts from this blog

Where can I find an analysis of Oliver Optic's Rich and Humble? (1863)

While there does not appear to be any substantial analysis online about Rich and Humble: Or The Mission of Bertha Grant by William Taylor Adams writing as Oliver Optic, at least one book has written about the novel and analyzed it a bit. One such book is In the Company of Books: Literature and Its Classes in Nineteenth-Century America by Sarah Wadsworth. Wadsworth discusses the novel in the context of the foreword written by Adams, saying the most interesting thing about the book is "the explicit claim it makes to target a female audience."  Wadsworth goes on to say that the novel does fulfill that description: it charts Bertha Grant's journey from luxury to poverty and back to luxury. However, she points out that James had written primarily for boys before Rich and Humble. There is also evidence in the text showing he wanted boys to have an entry point into the book—going so far as to address them in the foreword as well. She also points out that he includes multiple mal...
Read more

Why does Antonio say that he is ready to die?

In act 4, scene 1, Portia, disguised as a lawyer, leads Shylock and Antonio to believe that Shylock has the right to cut out a pound of flesh at the site of Antonio's heart, which, of course, will kill Antonio. It is a dramatic moment. Portia asks Shylock if he has the scale to weight the flesh he is about to cut. Portia then asks Antonio if he has any last words. Antonio, thinking he is about to die, uses his last words to comfort Bassanio. He loves Bassanio and does not want his friend to feel guilt that he might have caused Antonio's death by asking him for money. Antonio says he is ready to die because this is a way to spare Bassanio's feelings. Antonio wants to show Bassanio that he takes full responsibility for borrowing the money from Shylock. To further comfort Bassanio, Antonio says his death is really a blessing in disguise. If Shylock doesn't kill him, he will face his old age in poverty, which would be a terrible fate. As he puts it: Grieve not that I am fal...
Read more

What are the characters’ motivations? Are their motivations clear? If not, why are their motives elusive? How do certain characters act as literary foils to others? How do the characters develop or change in the story, and why?

Toward the end of Alice Walker’s “Everyday Use,” it becomes clear what each of the main characters' motivations are. In the beginning, Mama is excited to see her daughter Dee, who is a cosmopolitan, educated young woman who left behind her poor childhood home. Mama daydreams of being reunited with Dee on television in a dramatic fashion, which indicates that she hasn’t seen her daughter in some time—and she fantasizes about how the impending visit will go. Her other daughter, Maggie, is nervous about seeing her glamorous, more successful sister. When Dee almost demands to have precious family heirlooms, Mama realizes that Dee is condescending and unworthy of the objects she requests. When Mama refuses to give Dee the quilts she wants, Mama is motivated by a desire to see her family’s heritage honored rather than displayed in a false show of appreciation. Dee, on the other hand, is motivated by her desire to appear connected to her African ancestry and cultural heritage, though she ...
Read more