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 (limits-colimits) instead.

Filter by
Sorted by
Tagged with
0 votes
0 answers
29 views

Bounding a $\ln(x)/\ln(y)$ expression where L'Hopital's rule doesn't help

How might I upper bound the following within the unit interval as $p\to0$? $$ \frac{\ln(p)-\ln\left(\frac{x}{1-x}\right)}{\ln\left(1-\frac{(1-x)\ln(1-x)+x}{x^2}\right)} $$ The quotient is upper ...
user avatar
1 vote
0 answers
49 views

L'Hôpital's rule proof in Spivak Calculus

In chapter $11$ in Spivak Calculus he introduced L'Hôpital's rule that states that: $1$$-$ $\lim_{x \to a} f(x)=0$ $2-$ $\lim_{x \to a} g(x)=0 $ and if $\lim_{x \to a} \frac{f'(x)}{g'(x)}$ exists ...
user avatar
2 votes
1 answer
40 views

Help with a limit of a certain differential equation.

I have an ODE $y’+a(t)y=b(t)$ with initial condition $y(0)=0$. The functions $a(t),b(t)$ are continuous and each of them satisfy $$\lim_{t\rightarrow \infty}a(t)=A>0.$$ And $$\lim_{t\rightarrow \...
user avatar
  • 65
0 votes
0 answers
24 views

Why is it important that the argument of a function approaches an accumulating point, when defining a limit?

I'm starting to learn the fundamentals of calculus and I stumbled upon the definition of a limit. I don't understand exactly why the value the argument approaches must be an accumulating point of the ...
user avatar
  • 1
1 vote
1 answer
45 views

A question about double limits

For my project, I’m trying to prove the following claim: For all $\epsilon > 0$, $\exists X,Y \in S$ such that $\frac{F(X) + F(Y)}{F(X\cdot Y)} < 1 + \epsilon$. To keep things short, I might ...
user avatar
  • 11
4 votes
3 answers
43 views

Implicit Differentiation - $x^my^n = (x+y)^{m+n}$

Use implicit differentiation to find $\frac{\mbox{dy}}{\mbox{dx}}$ if $$x^my^n = (x+y)^{m+n}$$ Differentiation both sides with respect to $x$: $$mx^{m-1}y^n + x^mny^{n-1}y' =(m+n)(x+y)^{m+n-1}(1 + y')...
user avatar
  • 1,695
0 votes
1 answer
13 views

Existence (and calculation) of derivative for a piecewise function

During my Calculus course, my professor used a property to determine if a piecewise function is differentiable (and calculate the derivative of it) but it has never been proved. Here I attempt to ...
user avatar
  • 43
-1 votes
0 answers
61 views

Topological Limit of a function

$\newcommand{\dom}{\operatorname{dom}}$Lets say $B_{r}(x)$ is the family of balls of the point $x, r>0$. $ f$ is a real function $f:\Bbb R\to\Bbb R$. $ x$ is an accumulation point for the domain $\...
user avatar
  • 65
2 votes
1 answer
34 views

How to prove that $\lim_{p\rightarrow 0+} \|x\|_p=|\text{supp}\left(x\right)|$? [closed]

I have seen in many compressed sensing books (e.g. An Introduction to Compressed Sensing by Mathukumalli Vidyasagar) the following statement: $$\lim_{p\rightarrow 0+} \|x\|_p = \lim_{p\rightarrow 0+}...
user avatar
0 votes
1 answer
58 views

Finding all positive solutions to $\sqrt{x + 2\sqrt{x+\cdots + 2\sqrt{x+2\sqrt{3x}}}} = x$

Find all positive real solutions $x$ to the equation $$\sqrt{x + 2\sqrt{x+\cdots + 2\sqrt{x+2\sqrt{3x}}}} = x$$ and prove the expression is well-defined. The number of radicals is arbitrary. Clearly $...
user avatar
  • 741
5 votes
5 answers
112 views

Given that for each $n,\;x_n^n + x_n-1= 0,$ is $(x_n)_n$ convergent?

Prove that for $n\ge 2$, the equation $x^n + x-1 = 0$ has a unique root in $[0,1]$. If $x_n$ denotes this root, prove that $(x_n)_n$ is convergent and find its limit. The limit is $1$. But to find ...
user avatar
  • 741
0 votes
0 answers
27 views

What is one way to write the arc-length as a limit of a sum of lengths of line-segments?

I am trying to translate the following statement from Mathematical English into purely symbolic notation: Let $A$ be a pseudo-path. If $x$ is a point on path $P$ and and $d$ is a distance, then $A(x, ...
user avatar
-1 votes
2 answers
25 views

Error in Squeeze Theorem Argument

I am trying to write a proof using the squeeze theorem that $\lim\limits_{x \to 0} \frac{x}{\sin x} = 1$. My argument must be flawed because I keep coming up with $0$. The argument is as follows. We ...
user avatar
0 votes
0 answers
22 views

Number of discontinuous points related problem

If $f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } \left\{ {\begin{array}{*{20}{c}} {{{\left( {\cos x} \right)}^{2n}}}&{x < 0}\\ {{{\left( {1 + {x^n}} \right)}^{\frac{1}{n}}}}&{...
user avatar
0 votes
1 answer
31 views

Limit of partial sums where summands also depend on limit index

Let $a_i, a_i^n \in \mathbb{R}_{\geq 0}, i, n \in \mathbb{N}$ such that the series $\sum\limits_{i=1}^\infty a_i$ exists and $\lim\limits_{n \to \infty} a_i^n = a_i$ for all $i$. I would like to prove ...
user avatar
2 votes
2 answers
60 views

Find the limit of $\mathop {\lim }\limits_{x \to \infty } \left( {{x^p}\left( {\sqrt[3]{{x + 1}} + \sqrt[3]{{x - 1}} - 2\sqrt[3]{x}} \right)} \right)$

Value of p such that $\mathop {\lim }\limits_{x \to \infty } \left( {{x^p}\left( {\sqrt[3]{{x + 1}} + \sqrt[3]{{x - 1}} - 2\sqrt[3]{x}} \right)} \right)$ is some finite | non-zero number. My approach ...
user avatar
0 votes
2 answers
49 views

Differentiating $\sin x$ using limits

Below i am trying to differentiate $\sin x$ using first principle. $$\lim_{h \to 0} \frac{\sin(x+h)-\sin x}{h}$$ $$=\lim_{h \to 0} \frac{\sin x(\cos h-1)} {h}+\cos x\lim_{h \to 0} \frac{\sin h}{h}$$ ...
user avatar
  • 342
3 votes
3 answers
283 views

How to evaluate a limit that contains an integral?

This question is from the JEE Advanced 2007 Paper 1 (Question 49). The question states: $$\lim \limits_{x \to \frac{\pi}{4}} \frac{\int^{\sec^2x}_2 f(t)\ dt}{x^2-\frac{\pi^2}{16}}$$ I tried solving it ...
user avatar
1 vote
3 answers
73 views

Evaluating $\lim_{(x, y) \to (0, 0)} \frac{xy^2}{x^2-y^2}$ [duplicate]

What is the value of $$\lim_{(x, y) \to (0, 0)} \frac{xy^2}{x^2-y^2}$$ I have already tried the two paths result and I think now that the limit does exist and it is equals to 0. But i really cannot ...
user avatar
0 votes
0 answers
37 views

Conditional Fatou Lemma [closed]

Exercise If $M_t$ is a local martingale which is almost surely bounded from below by an integrable random variable $X$, show that $M_t$ is a supermartingale. Hint: apply Conditional Fatou Lemma. I ...
user avatar
10 votes
3 answers
158 views

Evaluate the limit $\lim\limits_{n\rightarrow \infty}\int_{0}^{\pi}\left|\sin(x)-\sin(2nx)\right|\text{ d}x$ [closed]

$$\lim_{n\rightarrow \infty}\int_{0}^{\pi}\left|\sin(x)-\sin(2nx)\right|\mathrm d x$$ Heading I put proper big numbers to $n$ using calculator to estimate the answer. And then I figured out that the ...
user avatar
2 votes
3 answers
84 views

Find the limit of $\frac{1+\frac13+\frac{1}{3^2}+...+\frac{1}{3^n}}{1+\frac15+\frac{1}{5^2}+...+\frac{1}{5^n}}$

Find the limit $$\lim_{n\to\infty}\dfrac{1+\frac13+\frac{1}{3^2}+...+\frac{1}{3^n}}{1+\frac15+\frac{1}{5^2}+...+\frac{1}{5^n}}$$ For the numerator we have $1+\frac13+\frac{1}{3^2}+...+\frac{1}{3^n}=\...
user avatar
  • 1,949
0 votes
2 answers
52 views

an infinitely differentiable monotonic function that goes to 0 at infinity but f' does not

We want infinitely differentiable monotonic function f such that: $\lim_{x \to +\infty} f(x) = 0$ and $\lim_{x \to +\infty} f'(x) \neq 0$. An infinitely differentiable bridging function, strictly ...
user avatar
-1 votes
1 answer
97 views

When is product of two functions equal to $0$?

We know that $$f(x)g(x)=0$$ if $f(x)=0$ or $g(x)=0$ Let $\epsilon>0$ be arbitrarily small. If we have $$\epsilon.f(x)=\epsilon.g(x)\tag{1}$$ then can we write $f(x)=g(x)$? But if we take $\epsilon\...
user avatar
4 votes
3 answers
146 views

Does Left Hand Derivate and Right Hand Derivative being defined guarantee continuity?

Suppose at $x = a$, both the Left Hand Derivative and Right Hand Derivative of a function exists and is defined. In other words, both the limits $$\lim_{h \to 0^-} \frac{f(a+h)-f(a)}{h}$$ and $$\lim_{...
user avatar
  • 1,695
0 votes
1 answer
45 views

Can we take individual derivative of piecewise function if the function is continuous and differentiable?

Can we take individual derivative of piecewise function if the function is continuous and differentiable? Suppose a function $f(x)$ is defined in such a way that it's definition changes at some ...
user avatar
  • 1,695
0 votes
1 answer
35 views

How can we remove the accuracy variable from this periodic real function?

Let $M(\alpha,\beta, \gamma)$ be the Euler Matrix and $f_A(\vec{v})$ the function that returns the azimuthal angle (spherical coordinate) corresponding to the Cartesian coordinates $(x,y,z)$ of the ...
user avatar
0 votes
0 answers
27 views

What is the „limit type” in the definition of Stieltjes-integral?

Let's say $f$ and $g$ are „nice enough” to $$\sum_{k}f\left(\xi_{k}\right)\left(g\left(x_{k+1}\right)-g\left(x_{k}\right)\right)\rightarrow\int_{a}^{b}f\left(x\right)dg\left(x\right),$$ where $\xi_{k}\...
user avatar
0 votes
0 answers
31 views

Proof for the multiplication law of limits

I am currently working on limits. In an early edition of "Calculus", Spivak uses the following lemma to proof the multiplication law of limits: Lemma: If $|x - x_0| < \min(1, \frac{\...
user avatar
1 vote
1 answer
29 views

Find all possible values of $a$ if $f(x) = \lfloor {\frac{(x-2)^3}{a}} \rfloor + a \cdot \cos(x - 2)$ is continuous in $[4, 6]$.

Find all possible values of $a$ if $$f(x) = \bigg\lfloor {\frac{(x-2)^3}{a}} \bigg \rfloor + a \cdot \cos(x - 2)$$ is continuous in $[4, 6]$. Hello, I think the answer to this problem is $\boxed{a \...
user avatar
  • 1,695
2 votes
1 answer
27 views

stricly convex function implication.

I'm work in the following statement Let $\Omega \subset \mathbb{R}^{n}$ be a open convex set and $f:\Omega\to \mathbb{R}$ a differential and strongly convex function in $\Omega$. Then for each $...
user avatar
2 votes
1 answer
82 views

Series expansion of $(1-cx)^{1/x}$

I am trying to understand the series expansion of $$(1-cx)^{1/x}$$ The wolframalpha seems to solve the problem by using taylor series for $ x\rightarrow 0$ and Puiseux series for $ x\rightarrow \infty$...
user avatar
  • 432
0 votes
1 answer
41 views

Inequalities with limits

Consider a function $f:\mathbb{R}\to\mathbb{R}$ that is continuous on $\mathbb{R}$ Suppose that $f(x) \geq 4\space\space\forall \space x\in [0,1)$ This implies that $$\lim_{x\to0^{+}}f(x)\geq 4$$ ...
user avatar
  • 109
1 vote
0 answers
32 views

If the determinant of a matrix goes to infinity, does the quadratic form of its inverse goes to zero? [closed]

This question occurs to me while I am self-studying machine learning. Let's assume there is a sequence of non-singular matrices $\{A_n\}$ whose size is the same and whose determinant goes to infinity: ...
user avatar
  • 33
9 votes
1 answer
185 views

Evaluate $\lim_{n \to \infty} \frac{a_n}{2 ^ {n - 1}}$ if $a_n = a_{n - 1} + \sqrt{a_{n - 1}^2 + 1}$

Let $a_i (i \in \mathbb{N}_{0})$ be a sequence of real numbers such that $a_0 = 0$ and $$a_n = a_{n - 1} + \sqrt{a_{n - 1}^2 + 1} \text{ } \forall n \geq 1$$ Evaluate the limit $$\lim_{n \to \infty} \...
user avatar
  • 1,695
0 votes
3 answers
72 views

How to compute $\lim_{x\to{\pi/2}} \frac{\cos{x} }{\sqrt[3]{(1-\sin{x})^2}}$ [closed]

$$\lim_{x\to{\pi/2}} \frac{\cos{x} }{\sqrt[3]{(1-\sin{x})^2}}$$ I am on terms with high school level calculus, and can solve this using L-hospital's method, but cannot come up with any other method to ...
user avatar
0 votes
1 answer
39 views

Let $f_n \to f$ pointwise. Are there $\varepsilon, N>0$ such that $|f_n (x)| \le |f (x)| + \varepsilon$ for all $n \ge N$ and $x \in X$?

Let $X \neq \emptyset$ and $f, f_n:X \to \mathbb R$ such that $f_n \to f$ pointwise, i.e., $f_n (x) \to f(x)$ for all $x \in X$. I would like to ask if any of below statements is correct. The meaning ...
user avatar
  • 11k
1 vote
1 answer
49 views

Let $p_n$ be the probability that the figure formed by connecting the four points $O,P,Q,R $ in this order is a convex square.

(Problem) Let rhombus OABC be a rhombus on the plane with one side of length $1$. $\angle A=60^{\circ} $. Let $n$ be a positive integer and $i, j, k $ are integers between $1$ and $n$. Three points P,...
user avatar
1 vote
1 answer
33 views

prove that $B(\mathbb{R}) = \mathcal{F}(L^1(\mathbb{R}))$

For $p=1,2,$ let $L^p(\mathbb{R})$ be the set of Lebesgue measurable functions $f:\mathbb{R}\to\mathbb{C}$ so that $\lVert f\rVert_p < \infty,$ where $\lVert f\rVert_1 := \int_{\mathbb{R}} |f(x)|dx$...
user avatar
  • 741
1 vote
1 answer
61 views

Is the statement $\sum_{j=1}^\infty x_j<\infty,~(x_j\ge0)$ $\Longrightarrow \lim _{k \to \infty} \sum_{j=k}^\infty x_j=0$ true?

As the title states, I would like to know if the statement $$ \sum_{j=1}^{\infty} x_{j}<\infty \Longrightarrow \lim _{k \to \infty} \sum_{j=k}^{\infty} x_{j}=0,\qquad x_j\in [0,\infty) $$ is always ...
user avatar
  • 549
3 votes
2 answers
81 views

Calculate $\lim_{n \to \infty} \left(\frac{n}{n^2 + 1} + \frac{n}{n^2 + 2} + \cdots + \frac{n}{n^2 + n}\right)$

Calculate $$\lim_{n \to \infty} \bigg(\frac{n}{n^2 + 1} + \frac{n}{n^2 + 2} + \cdots + \frac{n}{n^2 + n}\bigg)$$ Hello! The answer given to this problem is $1$, but I am getting to $0$: Consider $$\...
user avatar
  • 1,695
2 votes
3 answers
26 views

$2a-1< x+a < 2a+1 \implies |x+a| < 2|a|+1$

In tried to do a rigorous delta-epsilon argument for $\lim_{x \to a} x^2 = a^2$. I found this post, which I uses the same way as I did. But I don't fully understand on of the steps: If $|x-a| < 1$ ...
user avatar
0 votes
0 answers
32 views

How can prove limit of $ n\sin(\frac{1}{n})=1 $ as $ n $ approaches to infinity? [duplicate]

I am trying to solve this sequence limit $\lim_{n\to \infty} $ $n\sin(\frac{1}{n})=1$ and I’m struggling finding a bound for $|a_n -l|$ I thought of rewriting $|n\sin(\frac{1}{n}) - 1|$ As $|\frac {\...
user avatar
0 votes
4 answers
73 views

What is $\lim_{x \to 0} \frac{e^{x - \sin x} - 1}{x - \sin x}$ [closed]

Evaluate $$\lim_{x \to 0} \frac{e^{x - \sin x} - 1}{x - \sin x}$$ Hello! How do I find the limit without L'Hopital's rule? It is well known that $$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$$ So, if the ...
user avatar
  • 1,695
1 vote
1 answer
31 views

How to prove $f(x) = \frac{e^{x^{2}} \sqrt {\sin x}}{\cos x}$ is continuous on $[0,1]$ using limit theorems?

could you please give a hint for this? $$f(x) = \frac{e^{x^{2}} \sqrt {\sin x}}{\cos x}$$ I thought to approach breaking the function pieces by taking $e^{x^{2}} \sqrt{\sin x}$ as one function and $\...
user avatar
  • 35
0 votes
0 answers
20 views

Definition of sequentially compact subset

I am reading "Beginning Functional Analysis" by Karen Saxe, and I have came upon the definition of a sequentially compact subset: Let $(M,d)$ be a metric space. A subset $E \subset M$ is ...
user avatar
  • 151
0 votes
3 answers
91 views

Evaluating $\lim _{x\to \infty} \frac{\ln(2x+3)}{\sqrt[3]{x}}$ [closed]

Evaluate this limit $$\lim _{x\to \infty} \frac{\ln(2x+3)}{\sqrt[3]{x}}$$
user avatar
-8 votes
0 answers
37 views

Solving $\lim_{x\to1}\frac{(x+1)\sqrt{x^2-1}}{\sqrt{x^2+3x}-2}$ without the l'Hopital Rule [closed]

This is the limit I'm trying to solve but I'm unable to do so because I always arrive in a 0/0. $$\lim_{x\to1}\frac{(x+1)\sqrt{x^2-1}}{\sqrt{x^2+3x}-2}$$ Teacher says the solution is: ∞ Maybe she is ...
user avatar
7 votes
2 answers
146 views

show that if $\lim (f(x_1)+\cdots +f(x_n))/n$ exists whenever $\lim (x_1+\cdots +x_n)/n$ exists, then $f$ is continuous

Show that if $f:[0,1]\to \mathbb{R}$ and$$\lim \limits _{n\to \infty}\frac{f(x_1)+\cdots + f(x_n)}{n}$$exists whenever$$\lim \limits _{n\to \infty}\frac{x_1+\cdots + x_n}{n}$$exists, where $(x_n)$ is ...
user avatar
  • 741
-4 votes
1 answer
54 views

Is this function continuous [closed]

Suppose that $f: (0,1] → \mathbb{R}$ is such that $\lim \limits_{x \to a^-}f(x) =f(a)$ for all $a∈(0,1]$. Is $f$ continuous on $(0,1]$? I don't really know how to solve this question. I suspect that ...
user avatar
  • 23

1
2 3 4 5
808