# Tagged Questions

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### 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?
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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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### 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 ...
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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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### 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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### 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? ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 $\lim_{x\to-\infty}\ln(-x) = \infty$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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?
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### 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 ...
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### 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 ...
### 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.
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 ...