4
votes
4answers
303 views

Differentiation of $x^{\sqrt{x}}$, how?

The answer is (I think) $x^{\sqrt{x}-0.5} (1+0.5\ln(x))$, but how?
7
votes
2answers
624 views

When are the limit operations commutative?

I'll come up with a question that has bothered me for a long period of time. The question seems relatively simple, but I personally didn't manage to find an answer to it. In many cases I met problems ...
11
votes
3answers
2k views

$\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$

(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12) For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x > 0$. Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and ...
3
votes
2answers
942 views

Proof of an estimate for the tail of a normal distribution

My advisor told me to look up the proof of the following standard estimate so that we can adapt it to the case where we are dealing with something similar but including the addition of a polynomial ...
6
votes
2answers
306 views

Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials

Suppose that $$Q(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0} $$and $$P(x)=b_{m}x^{m}+b_{m-1}x^{m-1}+\cdots+b_{1}x+b_{0}.$$ How do I find $$\lim_{x\rightarrow\infty}\frac{Q(x)}{P(x)}$$ and what ...
30
votes
5answers
2k views

Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$

What methods can be used to evaluate the limit $$\lim_{x\rightarrow\infty} \sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x.$$ In other words, if I am given a polynomial $P(x)=x^n + a_{n-1}x^{n-1} ...
29
votes
2answers
1k views

Evaluating $\int P(\sin x, \cos x) \text{d}x$

Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$. For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$. Is there a general method which allows us to evaluate the ...
16
votes
3answers
2k views

Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$?

I was wondering if there is a name for a function that satisfies the conditions $f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \cdot f(y)$? Thanks and regards!
21
votes
3answers
1k views

Universal Chord Theorem

Let $f \in C[0,1]$ and $f(0)=f(1)$. How do we prove $\exists a \in [0,1/2]$ such that $f(a)=f(a+1/2)$? In fact, for every positive integer $n$, there is some $a$, such that $f(a) = ...
46
votes
15answers
3k views

Why does the series $\frac 1 1 + \frac 12 + \frac 13 + \cdots$ not converge?

Can someone give a simple explanation for why the harmonic series $$\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, but just grows very slowly? I'd prefer an easily ...