Tagged Questions

14
votes
5answers
1k 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?
0
votes
2answers
55 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
4answers
172 views

1 to the power of infinity, why is it indeterminate? [duplicate]

I've been taught that $1^\infty$ is undetermined case. Why is it so? Isn't $1*1*1...=1$ whatever times you would multiply it? So if you take a limit, say $\lim_{n\to\infty} 1^n$, doesn't it converge ...
2
votes
1answer
87 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
43 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
182 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 ...
10
votes
7answers
164 views

A proof of a property of limits

Today during lecture my lecturer showed us this property, but provided no proof. If $$\lim_{n\to\infty} {d_{n+1}\over d_n} >1$$ then $$\lim_{n\to\infty}d_{n}=\infty $$ Is this property legit? ...
1
vote
5answers
216 views

Evaluate $\lim\limits_{x \to \infty}\left (\sqrt{\frac{x^3}{x-1}}-x\right)$

Evaluate $$ \lim_{x \to \infty}\left (\sqrt{\frac{x^3}{x-1}}-x\right) $$ The answer is $\frac{1}{2}$, have no idea how to arrive at that.
2
votes
3answers
125 views

Could $\frac x0 = \pm\infty$? [duplicate]

Possible Duplicate: Is it wrong to tell children that 1/0 = NaN is incorrect, and should be ∞? I remember that dividing by zero is frowned upon, because it is said that there is no real ...
1
vote
4answers
263 views

Integration with infinity and exponential

How is $$\lim_{T\to\infty}\frac{1}T\int_{-T/2}^{T/2}e^{-2at}dt=\infty\;?$$ however my answer comes zero because putting limit in the expression, we get: $$\frac1\infty\left(-\frac1{2a}\right) ...
1
vote
1answer
74 views

Continuity and pushing a limit inside the function's domain

Consider some right-continuous function $f:\mathbb{R\cup\{-\infty,\infty\}}\to [0,1]$. I have to evaluate (i) $\lim_{b \to 0^+} f(\frac{a}{b})$, and (ii) $\lim_{b \to 0^-} f(\frac{a}{b})$ where $a \in ...
2
votes
3answers
130 views

Find: $\lim_{n\to\infty} r^n$, for $r>1$ and $r<1$

Prove: $$\lim_{n\to\infty}r^n = +\infty\,, r > 1;$$ $$\lim_{n\to\infty}r^n = 0\,, 0 \le r < 1.$$ I am not quite sure how to prove this, but once someone proves it I will make sure to ask ...
3
votes
4answers
131 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 ...
1
vote
3answers
72 views

What is positive-0 squared minus positive-0?

I've got a basic limit problem that I think I'm solving the right way, but I've run into something that looks confusing enough to make me wonder if I'm doing it right. $$ \lim_{y\to0} \frac{1}{y^2-y} ...
1
vote
1answer
108 views

The relative rates of tending towards infinity of different functions?

In my reading, it says that the function $x/\log x$ approaches infinity slower than $x$ (I got that bit), but then it says that it also approaches it faster that the functions $x^{1-d}$, where $d$ is ...
1
vote
4answers
152 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
90 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 ...
8
votes
3answers
517 views

Is the infinite root of any number equal to $1$?

I was messing around in IRB and I decided to make a $n^{th}$ root function and noticed that for very large roots of numbers, the answer always converges to $1$. It has been a while since I have done ...
15
votes
6answers
863 views

Limits of $f(x)=x-x$

It's obvious that $f(x)=x-x=0$. But what exactly happens here? You have a function $f(x)=x-x$ and you have to calculate the limits when $x\to \infty$ This'll be like this: $$\lim\limits_{x\to ...
2
votes
1answer
368 views

speed of convergence to infinity

Lets take for example $\lim_{x\rightarrow\infty} \log(x)$, from a mathematical point of view this is $+\infty$, but from a logical point of view it's clear that $x$ converges to $+\infty$ much more ...
1
vote
2answers
83 views

The limit of a rational function

$$ \lim _{x \to -\infty} \frac{3x^2+3x}{2x^2+2}$$ Is a good practice to do this? Change the $ -\infty $ to $ \infty $, and change the sign of the $ x $ variables: $$ \lim _{x \to \infty} ...
3
votes
3answers
134 views

Continuity of $f(x)$ involving infinity

$f(x)= \frac{\sin(\pi x)}{x(1-x)}$ How can I define $f(0)$ and $f(1)$ to make $f(x)$ continuous on $[0,1]$? I've found that the limit at $0 = \pi$, and the limit from the left at $1 = \infty$. I ...
99
votes
10answers
13k views

What is the result of infinity minus infinity?

What is $\infty - \infty$? Is it $\infty$ or $0$ or what?
5
votes
4answers
2k views

Negative 1 to the power of Infinity

Can anyone explain me what the result of $$\lim_{n\rightarrow\infty} (-1)^n$$ is and the reason?
5
votes
6answers
523 views

Why is $\infty^0$ indeterminate?

In a recent test question I was required to us L'Hopital's rule to evaluate: $$\lim_{x\to 0^+} x\ln{(e^{2x}-1)}$$ I assumed that anything multiplied by 0 would give an answer of 0. This turns out ...
1
vote
3answers
123 views

Regarding limits and $1^\infty$ [duplicate]

Possible Duplicate: Why is $1^{\infty}$ considered to be an indeterminate form I have some questions about limits and the undefinability of $1^\infty$. For example, is ...
2
votes
4answers
656 views

What is the limit as $x\to\infty$ of $\cos x$?

What is the limit as $x\to\infty$ of $\cos x$? Thanks in advance.
15
votes
3answers
1k views

On applying the quadratic formula to a first-degree equation

You're probably thinking, "Why?" Please let me explain... It is (very) well-known that $$ \forall (a,b,c,x) \in \mathbb{C}^* \times \mathbb{C}^3: ax^2 + bx + c = 0 \Leftrightarrow x = \frac{-b \pm ...