Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [limits]

Questions on the evaluation and properties of limits in the sense of analysis and related fields. For limits in the sense of category theory, use the tag “limits-colimits” instead.

In mathematics, a limit is the value that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

The formal $\varepsilon$-$\delta$ definition of a finite limit at a point $a\in \mathbb{R}$ is:

$$\Big(\lim_{x\rightarrow a} f(x) = L \Big)\iff \Big(\forall \varepsilon >0\, \exists \delta > 0: \forall x\in D\quad 0<\vert x-a\vert <\delta \implies \vert f(x)-L\vert <\varepsilon \Big).$$

The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to the concepts of limit and direct limit in category theory.

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Limit of $\sin\left(\frac{\pi}{x} \right)$ as $x$ approaches $0$ does not exist, with squeeze theorem

I understand that $\frac{\pi}{0}$ would be infinity, hence $\sin\left(\frac{\pi}{x} \right)$ does not have a limit as $x$ approach $0$. But if I would to use the squeeze theorem: $$-1 \leq \sin\left(\frac{\pi}{x} \right) \leq 1$$ shouldn't it be…
user316742
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What is the procedure to solve for the $\lim\limits_{x\rightarrow 0} \frac{\cos(3x)-1}{x^2}$?

What is the procedure to solve for the $\lim\limits_{x\rightarrow 0} \frac{\cos(3x)-1}{x^2}$? My calculator tells me the answer is -9/2, but I don't know how to solve without substituting values of x. I suspect there is some trigonometric identity…
CAGT
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Use the definition of convergence of a sequence to show $\lim \frac {2n^2}{n^3+3}= 0$

Use the definition of convergence of a sequence to show $$\lim_{n\rightarrow \infty} \frac {2n^2}{n^3+3}= 0$$ I understand that to do this we must show $ \frac {2n^2}{n^3+3} \leq \varepsilon$, but I'm not sure how to do that.
user312679
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Why is $\lim_{x\to 0^+} \left\lfloor\frac{\sin|x|}{x}\right\rfloor= 0\,?$

Problem: Check $$\lim_{x\to 0} \left\lfloor\frac{\sin|x|}{x}\right\rfloor$$ Now, what the book does is this: $$\textrm{RHL} \implies\lim_{x\to 0^+} \left\lfloor\frac{\sin|x|}{x}\right\rfloor = \lim_{h\to 0} \left\lfloor\frac{\sin|0…
user142971
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How would I find the limit for this equaiton?

How would you find the limit of this function: The n value is always even (this is for a problem and it would not make sense for n to be an odd number) Also, how could I plot this with n as the x value? I've tried to do it with an excel…
Marco
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Prove the following limit identity $\lim_{x\to\infty}\left( \csc\frac{m}{n+x}-\csc\frac{m}{x}\right)=\frac{n}{m}$

I am trying to prove the limit I came up with: $$\lim_{x\to\infty}\left(\csc\dfrac{m}{n+x}-\csc\dfrac{m}{x}\right)=\dfrac{n}{m}$$ This fact came from the double generalization of the special case $\,m=\pi,\;n=1\,$ which has exceptional connections…
user311151
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Asymptotic behavior of the expression: $(1-\frac{\ln n}{n})^n$ when $n\rightarrow\infty$

The well known results states that: $\lim_{n\rightarrow \infty}(1-\frac{c}{n})^n=(1/e)^c$ for any constant $c$. I need the following limit: $\lim_{n\rightarrow \infty}(1-\frac{\ln n}{n})^n$. Can I prove it in the following way? Let $x=\frac{n}{\ln…
Michael
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Confusion with a limit of Cothx

I am a Calc 2 student and am having trouble seeing this limit, the assignment is to use the definition of the hyperbolic functions to find the limit. $\lim_{x\to 0+} Coth(x)$ the answer is $+\infty$ If I work it out I get to this point: $lim_{x\…
Ryan
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Is there a proof for why $\lim_{x \to \infty} \frac{\log (x!)}{\log x^x} = 1$?

After playing around with the algebra, I found that the numerator became $\sum_{i}^{n} \log (i)$ while the denominator could be written as $\sum_i^n \log (n)$, but I couldn't get anywhere formally next. Informally, I see how something going from…
Austin
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Find the limit $\lim_{x \to 0} \left( 1 + \sin \left( \frac 3 x \right) \right)^x$

What is the answer to the limit? $$\lim_{x \to 0} \left( 1 + \sin \left( \frac 3 x \right) \right)^x$$ The book's answer shows that $e^3$ while I keep getting $e^0$. I used the estimation $$\lim_{x \to 0} x \ln x \le \lim_{x \to 0} x \ln \left( 1 +…
Mula Ko Saag
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Limit $\lim_{x \to 0 }\frac{x}{\sin x} = 1$?

I have a question regarding limits. Recently in a math class, my teacher states that $\frac{\sin x}{x}$ goes to $1$ hence in the case of a $\lim_{x\to 0} \frac{x}{\sin x}$, the answer is $1$. Why is that so? Shouldn't the answer be $0$ in this case?…
weejing
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Evaluating the limit $\lim_{x \to 1} (x^3 - 1) / (x - 1)$

$$\lim_{x \to 1} \frac{x^3 - 1}{x - 1}$$ As $x$ approaches to $1$, if I use the substitution method, it will become undefined. Then, I tried to multiply it by its conjugate but I still get undefined answer. How can I solve it?
user307537
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Am I using the term "limiting value" correctly?

Suppose "the limit as x goes to a of f(x) is L". I believe in school I had teachers used the term "limiting value" to refer to a. Is this the correct terminally to refer to a? I would like to use this in my class, but I want to make sure I'm…
B flat
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Finding limits by L'Hospital's Rule

The question reads as follows: Let $f(x) = x^{\frac{1}{x}}$ for $x>0$ Calculate $\lim_{x\to0^+} f(x)$ and $\lim_{x\to\infty} f(x)$ My attempt: First rewrite the function to a form where we can apply L'Hospital's Rule: $x^{\frac{1}{x}} =…
patrickh
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Find limit of the function $f(x) = \frac{\ln(x)}{\ln(2x)}$

Find limit of the function: $f(x) = \frac{\ln(x)}{\ln(2x)}$ Solve: $\lim\frac{\ln(x)}{\ln(2x)} =\lim\frac{(\ln(x))'}{(\ln((2x))'}=\frac{\frac{1}{x}}{\frac{1}{2x}} = 2 $ However, the answer is $1$. What did I do wrong?
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