# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 $x\in S$?
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### 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.
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### $\frac{1}{\infty}$ - 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? ...
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### 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?
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### 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.
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### Prove $\mathop {\lim }\limits_{x \to \pm \infty } {a \over x} = 0$ [closed]

How do you prove: $\mathop {\lim }\limits_{x \to \pm \infty } {a \over x} = 0$
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### Evaluation of a limit

Here is a question on limits. I would like to ask help. Here it goes: $$\lim_{N\to\infty}\left(\frac{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n+1}}{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n}}\right)$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $0$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\pi$ 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 ...
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### Limit approaching infinity-related question

Why is $$\lim_{x\to\infty}\frac{x^2}{1+x^2}=1?$$
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 $I$ ?
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### 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 ...
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### 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 ...