# Tagged Questions

Questions on the evaluation of limits.

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### What is the result of infinity minus infinity?

What is $\infty - \infty$? Is it $\infty$ or $0$ or what?
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### Why does factoring eliminate a hole in the limit?

$$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5} = \lim_{x\rightarrow 5} (x+5)$$ I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and ...
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I got stuck with a problem that pop up in my mind while learning limits. I am still a high school student. Define $P(m)$ to be the statement: $\quad ... 2answers 602 views ### How prove this nice limit$\lim_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$Nice problem: Let$a_{0}=1$and $$a_{n}=a_{\left\lfloor n/2\right\rfloor}+a_{\left\lfloor n/3 \right\rfloor}+a_{\left\lfloor n/6\right\rfloor}.$$ Show that ... 10answers 1k views ### What is the fastest/most efficient algorithm for estimating Euler's Constant$\gamma$? What is the fastest algorithm for estimating Euler's Constant$\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get$2$decimal ... 9answers 3k views ### How to prove that$\lim\limits_{x\to0}\frac{\sin x}x=1$? How can one prove the statement $$\lim\limits_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of$\sin$,$\cos$and$\tan$? Best would be a geometrical solution. This is homework. In my ... 9answers 3k views ### What is this beauty curve? Consider the following shape which is produced by dividing the line between$0$and$1$on$x$and$y$axes into$n=16$parts. Question 1: What is the curve$f$when$n\rightarrow \infty$? ... 7answers 2k views ### Evaluating$\lim_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using W|A, i may guess that the limit is$\frac{1}{2}$that is a pretty interesting and nice result. I ... 2answers 2k views ### Compute$ \lim\limits_{n \to \infty }\sin \sin \dots\sin n$I need your help with evaluating this limit: $$\lim_{n \to \infty }\underbrace{\sin \sin \dots\sin}_{\text{n compositions}} n,$$ i.e. we apply the$\sin$function$ntimes. Thank you. 8answers 3k views ### When to Stop Using L'Hôpital's Rule I don't understand something about L'Hôpital's rule. In this case: \begin{align} & {{}\phantom{=}}\lim_{x\to0}\frac{e^x-1-x^2}{x^4+x^3+x^2} \\[8pt] & ... 4answers 2k views ### Why does the google calculator give tan 90 degrees = 1.6331779e+16? I typed in tan 90 degrees in google and it gave 1.6331779e+16. How did it come to this answer? Limits? Some magic? 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.$$In other words, if I am given a polynomial P(x)=x^n + a_{n-1}x^{n-1} ... 2answers 615 views ### A beautiful limit involving primes and composites I observed the following limit empirically. Let p_n be the n-th prime and c_n be the n-th composite number then,$$ \lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^{n}\frac{p_n c_n}{p_n c_n + ... 4answers 1k views ### Find\lim\limits_{n \rightarrow \infty}\dfrac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}$Find$$\lim_{n \rightarrow \infty}\frac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}.$$ This is a recent exam question, which I couldn't figure out in the exam. My guess is it doesn't ... 2answers 551 views ### Computing$ \sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty} \frac{(-1)^{i+j}}{i+j}I would like to compute: $$\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{(-1)^{i+j}}{i+j}$$ I wanted to use Fubini's theorem for double series but $$... 8answers 1k views ### \lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2} I am able to evaluate the limit$$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$for a given n using l'Hôspital's (Bernoulli's) rule. The problem is I don't quite ... 3answers 1k views ### Why is this series of square root of twos equal \pi? Wikipedia claims this but only cites an offline proof:$$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+... \sqrt 2}} = \pi$$for n square roots and one minus sign. The formula is not the "usual" one, like ... 6answers 1k views ### Uses of \lim \limits_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} I have been wondering whether the following limit is being used somehow, as a variation of the derivative:$$\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} .$$Edit: I know that this limit is defined in ... 4answers 425 views ### \lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} n^{1/k} What would you suggest here?$$\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} n^{1/k} $$4answers 658 views ### Limit of \log (\log( … \log((n)^ {(n-1)^ {…}}))) This is a spinoff of this question Defining$$f_0(x) = xf_n(x) = \log(f_{(n-1)} (x)) \space (\forall n>0)$$and$$a_0 = 1a_{n+1} = (n+1)^{a_n} \space (\forall n>0)$$How to ... 3answers 1k views ### Find the limit \lim \limits_{n\to \infty }\cos \left(\pi\sqrt{n^{2}-n} \right) I'd love your help with finding the following limit:$$\lim_{n\to \infty }\cos (\pi\sqrt{n^{2}-n}).$$I was asked to find this limit, but honestly I believe that it doesn't exist. According to Heine ... 8answers 2k views ### When two functions are equal, but not. I haven't looked into it much, but this is something I've been aware of that I know I need to look into. When I have a function f(x)=\frac{x+1}{x+1}, There is a discontinuity at x=-1, yet ... 1answer 418 views ### Power towers: to infinity and all the way back In the following, let n be a positive integer, all other variables be real (furthermore, a>1), all functions be real-valued, and logarithms of negative arguments be undefined. Let ... 6answers 1k views ### Proofs of \lim\limits_{n \to \infty} \left(H_n - 2^{-n} \sum\limits_{k=1}^n \binom{n}{k} H_k\right) = \log 2 Let H_n denote the nth harmonic number; i.e., H_n = \sum\limits_{i=1}^n \frac{1}{i}. I've got a couple of proofs of the following limiting expression, which I don't think is that well-known: ... 4answers 6k views ### When can you switch the order of limits? Suppose you have a double sequence \displaystyle a_{nm}. What are sufficient conditions for you to be able to say that \displaystyle \lim_{n\to \infty}\,\lim_{m\to \infty}{a_{nm}} = \lim_{m\to ... 2answers 540 views ### Compute \lim\limits_{n\to\infty} \prod\limits_2^n \left(1-\frac1{k^3}\right) I've just worked out the limit \lim\limits_{n\to\infty} \prod\limits_{2}^{n} \left(1-\frac{1}{k^2}\right) that is simply solved, and the result is \frac{1}{2}. After that, I thought of calculating ... 1answer 664 views ### Repeated Factorials and Repeated Square Rooting I was talking with friends about silly questions involving what numbers you can get using only a single digit "3" and unary operations. We eventually conjectured that using only factorials and square ... 6answers 2k views ### Which infinity is meant in limits? For example, when we write \lim_{x\rightarrow \infty} f(x) - which infinity is meant and why? Countable? If uncountable - which and why? 4answers 1k views ### \sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}} approximation Is there any trick to evaluate this or this is an approximation, I mean I am not allowed to use calculator.$$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$$8answers 595 views ### Infinite powering by i [duplicate] Find the value of: i^{i^{i^{i^{i^{i^{....\infty}}}}}} Simply infinite powering by i's and the limiting value. Thank you for the help. 4answers 788 views ### How can I show that \sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}} exists? I would like to investigate the convergence of$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt\ldots}}}}$$Or more precisely, let$$\begin{align} a_1 & = \sqrt 1\\ a_2 & = \sqrt{1+\sqrt2}\\ a_3 ... 2answers 483 views ### Crafty solutions to the following limit The following problem came up at dinner, I know some ways to solve it but they are quite ugly and as some wise man said: There is no place in the world for ugly mathematics. These methods are using ... 3answers 611 views ### When L'Hôpital's Rule Fails I was discussing L'Hôpital's Rule with a Calculus I student earlier today. I mentioned that if the limit obtained by differentiating the numerator and denominator doesn't exist, then L'Hôpital's Rule ... 5answers 404 views ### How to find\lim_{n\to\infty}\frac{1!+2!+\cdots+n!}{n!}$? How to evaluate the following limit? $$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}$$ For this problem I have two methods. But I'd like to know if there are better methods. My solution 1: Using ... 1answer 529 views ### Convergence of the series$\sum_{n=1}^\infty \frac{(\sin n)^n}{n}$. Please determine whether the series$\displaystyle\sum_{n=1}^\infty \frac{(\sin n)^n}{n}$converges. (Note: In Mathematica, the result tends to converge. Moreover, this is a problem mis-copied from ... 2answers 192 views ### On the Paris constant and$\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\dots}}}}$? In 1987, R. Paris proved that the nested radical expression for$\phi$, $$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$$ approaches$\phi$at a constant rate. For example, defining$\phi_n$as ... 12answers 2k views ###$ \lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$is infinite How do I prove that$ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$is infinite? 1answer 1k views ### What is the limit of$n \sin (2 \pi \cdot e \cdot n!)$as$n$goes to infinity? I tried and got this $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where$m$is an ... 4answers 503 views ### Limit of series involving ratio of two factorials $$\sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3}$$ The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ... 2answers 501 views ### Evaluating the infinite series$\sum\limits_{n=1}^\infty(\sin\frac1{2n}-\sin\frac1{2n+1})$I've been bored and playing with infinite series and came across in my book the following problem, namely to determine the convergence ... 7answers 1k views ### Is$\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$? Is$\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$I.e - does the second half of the harmonic series go to zero? I know that for a finite number of terms the limit of the sum is ... 3answers 941 views ###$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$Evaluating this limit Evaluate $$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$$ 2answers 410 views ###$\lfloor0.999\dots\rfloor= ?0$or$1$? I think$\lfloor0.999\dots\rfloor= 1$, as$0.999\dots=1$,but I have doubt, as$\lfloor0.9\rfloor=0$,$\lfloor0.99\rfloor=0$,$\lfloor0.9999999\rfloor=0$, etc. 4answers 544 views ### Computing$\lim\limits_{n\to+\infty}n\int_{0}^{\pi/2}xf(x)\cos ^n xdx$I got stuck at the following problem. Let$f\in C([0,\pi/2])$, then compute $$\lim_{n\to+\infty}n\int\limits_{0}^{\pi/2}xf(x)\cos ^n xdx$$ Could you suggest a helpful idea? 4answers 2k views ### How to show that$\lim\limits_{x \to \infty} f'(x) = 0$implies$\lim\limits_{x \to \infty} \frac{f(x)}{x} = 0$? I was trying to work out a problem I found online. Here is the problem statement: Let$f(x)$be continuously differentiable on$(0, \infty)$and suppose$\lim\limits_{x \to \infty} f'(x) = 0$. ... 2answers 477 views ### Calculating$\lim_{n\to\infty}\sqrt{n}\sin(\sin…(\sin(x)..)$I was asked today by a friend to calculate a limit and I am having trouble with the question. Denote$\sin_{1}:=\sin$and for$n>1$define$\sin_{n}=\sin(\sin_{n-1})\$. Calculate ...
What is the limit of the continued fraction $$\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{7+\cfrac{1}{11+\cfrac{1}{13+\cdots}}}}}}\ ?$$ Is the limit algebraic, or expressible in terms of e or ...