Articles of calculus

asymptote problem?

I was curious about this exercise, because I thought it could be a valuable tool to use the theorem dell’asintoto … I do not think that is the way, does anyone have any idea? ! Let $ h $ is a function defined on $(a, + \infty)$ and limited all intervals $(a, b)$ $a <$ […]

Computing a Double Limit

How would one compute $\lim_{\delta \rightarrow 0, k\rightarrow\infty} (1+\delta)^{ak}$, where $a$ is some positive constant? I am finding a lower-bound of the Hausdorff Dimension on a Cantor-like set and this expression appeared in my formula. Here’s what I have, even though I’m not sure if I can use L’Hopital in this case (where $k, \delta$ […]

Lacking information on a Calculus Problem, have $f(x)$ need ${{dg} \over {dx}}$?

I am a senior in high school taking an AP Calc AB class. The textbook I am using is Calculus Concepts and Calculators (second edition) distributed by Venture Publishing. I am utterly stuck on question 18. Let f(x) = int(x). a) Explain why g’(2.5) = 0. b) Explain why g’(2) does not exist. I am […]

What's $\limsup_{(h_x,h_y)\to(0,0)} \frac{\left|\frac{x+h_x}{y+h_y}-\frac{x}y\right|} {\sqrt{{h_x}^2+{h_y}^2}}$?

This originally comes from $f_1(x,y)=\frac{x}{y}$, where $X=\mathbb{R}^{n}, Y=\mathbb{R}^m, x \in X, f: X \rightarrow Y, x \neq 0, f(x) \neq 0$ $$\limsup_{(h_x,h_y)\to(0,0)} \frac{\left|\frac{x+h_x}{y+h_y}-\frac{x}y\right|} {\sqrt{{h_x}^2+{h_y}^2}}$$ If I understand it correctly, $x$ and $y$ can be anything but zero, and $h_x,h_y$ go towards zero. Moreover, both numerator and denominator cannot be negative. But since $x,y$ could be […]

Absolute and Conditional Convergence of the integral $\frac{\sin(x)}{x^p}$ for real values of $p$

This question already has an answer here: Convergence $I=\int_0^\infty \frac{\sin x}{x^s}dx$ 5 answers

Differentiate the Function: $y=x^x$

$y=x^x$ Use $\frac{d}{dx}(a^x)=a^x \ln a$ My answer is: $x^x \ln x$ The book has the answer as $x^x\ (1+ \ln\ x)$ Am I missing a step?

How to find this limit with following constraints?

Question: If a,b are 2 positive , co-prime integers such that $$\lim _{n \rightarrow \infty}(\frac{^{3n}C_n}{^{2n}C_n})^\frac{1}{n}=\frac{a}{b}$$Then a+b?? I tried to break down the limit to: $$\frac{[(3n)(3n-1)\dot{}\dot{}\dot{})^\frac{1}{n}][n(n-1)(n-2)\dot{}\dot{}\dot{})^\frac{1}{n}]}{[2n(2n-1)(2n-2)\dot{}\dot{}\dot{}]^\frac{2}{n}}$$ but I’m lost ahead of it. Answer given in my text books is 43. Please guide me

Prove the function $f(x)=1-x-(1-\frac{a}{n}x)^n$ has a real solution between $(0,1]$

Consider the function $f(x)=1-x-(1-\frac{a}{n}x)^n$ where $a$ and $n$ are both constants such that $\frac{a}{n}\in[0,1]$ ( it represents some probability indeed). One obvious solution is $x=0$. I want to prove that such a function has a real solution between $(0,1]$ and this solution has the form that dependening on $a$ and independent of $n$. The only […]

Two simple series

I dont know how to calculate these two series: $$\begin{align} & \sum\limits_{n=1}^{\infty }{\frac{n+3}{{{n}^{3}}+\ln n}} \\ & \sum\limits_{n=1}^{\infty }{\left( 1-\cos \frac{\pi }{n} \right)} \\ \end{align}$$

Finding area of a triangle with integrals

I am suppose to find the area of a triangle using integrals with vertices 0,0 1,2 and 3,1 This gives me $y= 2x$ $y=\frac{1}{3}x$ $y= \frac{-1}{2}x+\frac{5}{2}$ for my slopes I know that I can calculate the area of the first part by finding $$\int_{0}^{1}2x-\frac{1}{3}x$$ and the second part by $$\int_{1}^{3}\frac{-1}{2}x+\frac{5}{2}-\frac{1}{3}x$$ The anti derivative of the […]