1
vote
4answers
90 views

Evaluate the limit: $\lim_{x\to \infty}$

Evaluate the limit: $$\lim_{x\to\infty} \frac{(2x^2 +1)^2}{(x-1)^2(x^2+x)}$$ The answer is 4 and I don't understand why, but why can't I just do something like:$$\frac{(\infty)}{(\infty)(\infty)} = ...
3
votes
2answers
102 views

I need to understand why the limit of $x\cdot \sin (1/x)$ as $x$ tends to infinity is 1

here's the question, how can I solve this: $$\lim_{x \rightarrow \infty} x\sin (1/x) $$ Now, from textbooks I know it is possible to use the following substitution $x=1/t$, then, the ecuation is ...
1
vote
1answer
58 views

Assigning values to divergent integrals

I'm interested in the (obviously divergent) integral $$ \int_{-\infty}^\infty dx e^{-x f}\ ,$$ where $f$ is real. Is there any way to meaningfully assign a value to this integral? I was thinking of ...
1
vote
2answers
60 views

Is the inverse ackermann function the slowest growing function that goes to infinity?

Actually, this is not precisely my question. If $a(x)$ is the inverse ackermann function, then obviously $a(a(x))$ grows slower than $a(x)$, as does $\log(a(x))$, and so on. But is there a function f ...
2
votes
2answers
57 views

Doubts about infinite nested root

Find $f(a)=\sqrt{a-\sqrt{a^2-\sqrt{a^4-\cdots}}}$ where $a\in\mathbb{R}$. My Attempt : I consider $\frac{f(a)}{a}=\sqrt{1-\sqrt{1-\sqrt{1-\cdots}}}$. Now to finding this limit is easy but I cannot ...
3
votes
4answers
84 views

Calculate $\displaystyle \lim_{x \to \infty} x - \sqrt{x^2 + 2x}$ without derivations.

How can I calculte $\displaystyle \lim_{x \to \infty} x - \sqrt{x^2 + 2x}$? Here is what I´ve done so far: Multiplying by $\displaystyle \frac{x + \sqrt{x^2 + 2x}}{x + \sqrt{x^2 + 2x}}$ I got ...
2
votes
8answers
272 views

How come $1^{\infty}$ = undefined, while $2^{\infty} = \infty$ and $0^{\infty} = 0$? [duplicate]

$1^\infty$ = undefined $2^\infty = \infty$ $0^\infty = 0$ Why is $1^\infty$ undefined? People were trying to explain to me that infinity isnt part of the Real numbers, yet, $2^\infty$ and ...
2
votes
4answers
141 views

What is the answer to the paradox of the infinitesimal?

I just read this article on npr, which mentioned the following question: You can keep on dividing forever, so every line has an infinite amount of parts. But how long are those parts? If they're ...
0
votes
1answer
59 views

Calculus 2 Integral Question

I've been trying to resolve a calculus question and seem to be having troubles understanding exactly how to approach it. Some hints are supplied, but they don't exactly seem to help. Thanks to anyone ...
8
votes
4answers
436 views

Why does $ (\frac{1}{2})^∞ = 0?$

Recently while at my tutoring I had a question that said: "Aladin has a pair of magic scissors that can cut things in to tiny pieces. If he cuts a carpet in half, cuts the half into half and continues ...
0
votes
2answers
61 views

Finding a limit with a Square Root

$$\lim_{x\to \infty} \frac{\sqrt{9x^6-x}}{x^3+7}$$ I thought it would simply be $1/3$, not sure where I went wrong.
0
votes
1answer
136 views

Finding limit of a function as it approaches infinity

How do i solve the below without using L'hopital rule. The final answer obtained is $2/3$ ...
30
votes
6answers
2k views

Why is there antagonism towards extended real numbers?

In my backstory, I was introduced to the geometric concept of infinity rather young, through reading about the inversive plane. In the course of learning calculus, I'm pretty sure I formed a concept ...
0
votes
2answers
69 views

How to show by the Root Test that $\sum\limits_{i=1}^\infty (2n^{1/n}+1)^n$ converges or diverges

How do I show by the Root Test that $$\sum\limits_{i=1}^\infty (2n^{1/n}+1)^n$$ converges or diverges? This is what I have done so far. Since we take $\sum\limits_{i=1}^\infty \sqrt[n]{|a_n|}$, we ...
0
votes
2answers
65 views

Evaluating a limit with infinity

I'm taking the limit as x approaches infinity from the left (-) of: $$ \sqrt{x^2+2x}- \sqrt{x^2-2x} $$ However I'm not sure how to go about this. I'm at: $$ \sqrt{ \frac{x^3+4x^2}{x+2x}}- \sqrt ...
2
votes
2answers
48 views

Finding asymptotes of $(2-x^2)/(e^x)$

I was trying to solve some exam question on calculus 1, and i found this "Sketch the graph of $(2-x^2)/(e^x)$" I'm interested to find Horizontal Asymptotes of the graph. 1) when x approaches ...
6
votes
3answers
699 views

Limits at infinity

I'm working with limits at infinity and stumbled upon this exercise where I want to evaluate the indicated limit: $$\lim_{x \to \infty} \frac{1}{\sqrt{x^2-2x}-x}$$ I tried to solve it by doing the ...
0
votes
0answers
42 views

A simple question on limits

Is it true that $$ \lim_{x\to+\infty} \mathbb{I}_{S=\{z\mid e^{-z}>0, z\in\mathbb{R}\}}(x) = 1,$$ where $\mathbb{I}_{S}(x)$ is an indicator function for $x\in S$?
3
votes
4answers
87 views

Is it valid to write $\displaystyle \lim_{x \to 0} \frac{1}{x^2} = \infty$?

AFAIK the limes of a term does not exist if that term does not converge, but I haven't found a suiting question here yet. This probably is a double of a similar question.
6
votes
3answers
282 views

$\frac{1}{\infty}$ - is this equal $0$? [duplicate]

I've seen that wolfram alpha says: $$\frac{1}{\infty} = 0$$ Well, I'm sure that: $$\lim_{x\to \infty}\frac{1}{x} = 0$$ But does $\frac{1}{\infty}$ only makes sense when we calculate it's limit? ...
-2
votes
3answers
50 views

if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ $(A_n+B_n) \Longrightarrow \infty$

if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ $(A_n+B_n) \longrightarrow \infty$ How do you prove it?
1
vote
0answers
55 views

Prove infinity arithmatics

How do you prove $ \infty * (-\infty) = -\infty$ or $ \infty +\infty = \infty$? I thought it is an axiom, but have been there's is proof for that.
0
votes
1answer
49 views

Prove $\mathop {\lim }\limits_{x \to \pm \infty } {a \over x} = 0$ [closed]

How do you prove: $\mathop {\lim }\limits_{x \to \pm \infty } {a \over x} = 0$
0
votes
1answer
46 views

Evaluation of a limit

Here is a question on limits. I would like to ask help. Here it goes: $$\lim_{N\to\infty}\left(\frac{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n+1}}{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n}}\right)$$ ...
1
vote
2answers
651 views

Limits of trigonometric functions as $x$ approaches $\infty$

A while back I ran into a problem in which I had to analyze the graph of $f(x) = ( \arctan x )^2$. I was fine until I had to evaluate the limit of the function as is approaches infinity to determine ...
0
votes
4answers
306 views

Why does Wolfram Alpha state that $-\infty/0 = +\infty$?

I ran into a scenario when practicing L'Hôpital's rule which yielded -infinity/0. I broke this down into $-1 \cdot \infty \cdot \frac 1 0$, which I assumed equaled $-1\cdot\infty\cdot\infty$, which ...
4
votes
1answer
322 views

Hilarious Comic … DiffyQ and infinity ensue…

I ran across this comic, and it's gold. It is orginially published here If I am correct, the first panel alone defines a self-referential loop if not a differential Equation: $X$: Amount of Black ...
10
votes
6answers
548 views

Difference between approaching and being exactly a number

When we take a limit, we say that the value is never equals that number, but approaches it, like in $$\lim_{n\to\infty}\frac{1}{n} = 0.$$ It never reaches $0$, but becomes closer and closer to $0$. ...
0
votes
2answers
275 views

Question on limits and infinity

Just to clarify, the limit of $x \nearrow 0$ from the left of $1/x$, would be $-\infty$, and the limit of $x \searrow 0$ from the right of $1/x$, would be $+\infty$ right? This is only true when its ...
3
votes
2answers
119 views

How can evaluating the limit of function give a different result after rationalizing it?

One of the examples in Calculus: A complete course is finding $\lim_{x\to \infty} (\sqrt{x^2+x}-x)$. At first it seems to produce a meningless $\infty-\infty$, but by rationalizing it we eventually ...
4
votes
1answer
159 views

Can a curve be an asymptote?

$f(x)=x^3+\frac{3}{x-1}$ This was the question given to me.I replied that $f(x)$ will have only a single vertical asymptote of $x=1$. My teacher told that there'll be be two asymptotes.One is the ...
1
vote
1answer
107 views

$\pi$ and $\ln4$ relations. Even and Odd alternating sums.

Tonight, playing around on WolframAlpha, I discovered that the alternating sum of the odd numbers is $\frac\pi4$ and the alternating sum of the even numbers is $\frac{\ln4}4$ Are there any known ...
0
votes
2answers
73 views

Limit approaching infinity-related question

Why is $$\lim_{x\to\infty}\frac{x^2}{1+x^2}=1?$$
3
votes
2answers
241 views

Proof of a Property of Vertical Asymptotes

I'm trying to understand a proof in my Calculus textbook of the following theorem: Let the functions $f$ and $g$ be continuous on an interval containing $c$. If $f(c) \neq 0$, $g(c) = 0$, and ...
3
votes
2answers
149 views

Are real numbers also hyperreal? Are there hyperreal $\epsilon$ between $-a$ and $a$ for any positive real $a$?

The set of all hyper-real numbers is denoted by $R^*$. Every real number is a member of $R^*$, but $R^*$ has other elements too. The infinitesimals in $R^*$ are of three kinds: positive, negative ...
1
vote
3answers
585 views

infinity times infinitesimal - what happens?

So what happens if we multiply infinite number by. Infinitesimal number? Like $dx \times \infty$ where $dx$ is treated as in one-dimensional integration. Also, can we divide infinite number by ...
0
votes
2answers
137 views

Understanding limits at infinity with regard to the definition of a limit

This is sort of a follow up to my previous question Say you have $$ \lim_{x\to +\infty} f(x) $$ where $f : \mathbb{R} \to \mathbb{R} , x \in \mathbb{R}$ What exactly does this mean? From the last ...
2
votes
1answer
933 views

Infinity = Undefined?

Let's start with the equation $y = |1/(x-1)|$. The positive and negative limit of $x$ at $1$ both approach $+∞$, but at $x = 1$, $y$ is undefined. I know this is because the denominator of the ...
0
votes
1answer
86 views

Calculus Limit -> inf Question. Kindly Explain the First Step, encircled in red color.

Link to view Solution of my question in image format: Solution is correct. Kindly Explain the First Step, encircled in red color.
0
votes
1answer
53 views

Comparing Improper Integrals Involving Infinity

From my current understanding: $K>J$ and $L>K$ , therefore $L>K>J$. How can I compare the first integral $I$ ?
0
votes
1answer
577 views

Comparison Theorem for Integral Calculus

I have narrowed it down to C, E, and F, since we know that $1/x^{1/5}$ is always greater than the original function for all $x\geq 1$. However, the second set of conditions is more difficult to ...
0
votes
1answer
99 views

Identifying an Error in Determining the Convergency of an Infinite Series

Given the infinite series of $(-1)^n/(nln(n))$ for $n = 2,3,4,\ldots$ to infinity, is the series conditionally convergent, absoultely convergent, or divergent? I took two approaches to solve this ...
1
vote
0answers
70 views

Simplifying this infinite series [duplicate]

Possible Duplicate: How can I evaluate $\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$ I have an infinite series like so: $$\sum_{i=0}^\infty (i+1)x^i$$ or basically $$ 1 + 2x + 3x^2 + 4x^3 +... ...
3
votes
4answers
236 views

Limit of difference of two irrational functions

Firstly, this is not a homework. I just want to solve this limit for my own curiosity and self-learning. I have tried to solve this limit for 5-6 hours with no luck. Then I tried to read information ...
4
votes
2answers
34k views

Whats infinity divided by infinity?

This should be a simple question but i just want to make sure. I know from infinity/infinity is undefined. However if we have 2 equal infinities divided by each other it would be 1? And if we have ...
1
vote
7answers
3k views

Why do we say the harmonic series is divergent? [duplicate]

If we have $\Sigma\frac{1}{n}$, why do we say it is divergent? Yes, it is constantly increasing, but after a certain point, $n$ will be so large that we will be certain of millions of digits. If we ...
1
vote
4answers
176 views

why $\lim_{x\to-\infty}(\sin x+2)\ln(-x)=\infty$?

Why does $\lim_{x\to-\infty}(\sin x+2)\ln(-x)$ equal $\infty$? Breaking up the limit: $\lim_{x\to-\infty}(\sin x+2)$ DNE because it oscillates between 1 and 3 $\lim_{x\to-\infty}\ln(-x) = \infty$ ...
2
votes
2answers
110 views

Looking for some function such that $\lim\limits_{x\to\infty}f(x) \ne \infty$

I am looking for a function $f$ that is differentiable and $f'(x) \ge c \gt 0$ for all $x \in \mathbb{R}$ and $\lim\limits_{x\to\infty}f(x) \ne \infty$? Is there such function, or am I wasting my ...
3
votes
1answer
861 views

Why do some divergent series apparently seem to converge (e.g. Grandi's series)?

Grandi's series is defined as: $$\sum_{n=0}^{\infty} (-1)^n = 1 - 1+1-1+\cdots$$ By plainly looking at this series it seems like the value of it is either $1$ or $0$ by doing the following ...
7
votes
1answer
445 views

Am I thinking about infinitesimals correctly?

I was all set to begin Calculus 2 when I thought, "I should have a more intuitive sense of what's happening with differentials before I move on." I want to tell you what I've learned and ask you all ...