Questions on the evaluation and properties of limits.

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-3
votes
5answers
59 views

Evaluate $ \lim_{ x \to 0}{ \frac{e^{e/x}-e^{-e/x}}{e^{1/x}+e^{-1/x}} }$ [on hold]

How to evaluate limit of the following function at x=0 ? $$ \lim_{ x \to 0}{ \frac{e^{e/x}-e^{-e/x}}{e^{1/x}+e^{-1/x}} } $$
3
votes
1answer
34 views

If ${a_i} \to 0$ and $\{ {X_i}\} _{i = 1}^\infty $ is a sequence of iid random variables with zero mean, does ${a_i}{X_i} \to 0$ almost surely?

My problem is slightly more specific than the title of this question: Let $0 < \beta < 1$ and let $\{ {X_i}\} _{i = 1}^\infty $ be a sequence of i.i.d. random variables with $E({X_i}) = 0$. In ...
0
votes
0answers
16 views

Rankine Hugoniot, taking limits

I have seen two different derivations of the Rankine Hugoniot jump conditions across a shock s(t) in the xt-plane. I present a summary of the two different derivations and then post my question in ...
0
votes
0answers
23 views

Yet another asymptotic series that needs to be analyticaly extended

Let $A>0$ and $1\le \mu \le 2$. Consider a following definite integral: \begin{equation} {\mathcal I}(A,\mu) := Re\left[\int\limits_0^\infty e^{-(k A)^\mu}\frac{\left(\gamma+\Gamma(0,\imath ...
-1
votes
1answer
55 views

What is the limit of this product? (SOLVED)

What does this limit equal? $$\lim\limits_{k\to\infty}\left(\prod_{n=1}^kn^{2^{k-n}}\right)^{\frac{1}{2^{k-1}-1}}$$ All that I have tried so far is computation and it does seem to converge. I ...
3
votes
0answers
50 views

List of techniques to evaluate limits?

I'd like to make a complete list of techniques to solve a limit. Definition of the limit Continuous functions Algebra of limits Addition, multiplication, division Composition Inverse function ...
-1
votes
2answers
28 views

What is the difference of the greatest of the limits $\overline{\lim}_{n\to \infty}$ and the least of the limits $\underline{\lim}_{n\to \infty}$?? [on hold]

What are exactly these three limits for an infinite series $x_n$? $$\overline{\lim_{n\to \infty}} x_n$$ $$\underline{\lim}_{n\to \infty} x_n$$ $$\lim_{n\to \infty} x_n$$ Can they be different from ...
2
votes
4answers
56 views

Why $\lim_{\Delta x\to 0} \cfrac{\int_{x}^{x+\Delta x}f(u) du}{\Delta x}=\cfrac{f(x)\Delta x}{\Delta x}$?

I'm reading Nahin's: Inside Interesting Integrals. I've been able to follow it until: $$\lim_{\Delta x\to 0} \cfrac{\int_{x}^{x+\Delta x}f(u) du}{\Delta x}=\cfrac{f(x)\Delta x}{\Delta x}$$ I ...
2
votes
1answer
60 views

Chain Rule For Limits

Given a function $f:\mathbb R^2 \rightarrow \mathbb R$ is continuous and has a limit at $p=\infty$, $\lim_{x\rightarrow p}f(x,y)=b(y)$ and a function $g:\mathbb R \rightarrow \mathbb R$ is continuous ...
6
votes
1answer
41 views

Limit behavior of a definite integral that depends on a parameter.

Let $A>0$ and $1\le \mu \le 2$. Consider a following integral. \begin{equation} {\mathcal I}(A,\mu) := \int\limits_0^\infty e^{-(k A)^\mu} \cdot \frac{\cos(k)-1}{k} dk \end{equation} By ...
19
votes
3answers
155 views

Evaluate $\displaystyle\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$

Evaluate $$\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$$ I simplified the limit to $$\dfrac{1}{2}\lim_{n \to \infty} \int_{0}^{\frac{1}{2}} ...
0
votes
1answer
25 views

Compute the definite integral of f(x) as a limit of Riemann sums.

$[-1, 0] = [a,b]$ and $f(x) = 4x-1$. When I attempt to solve this problem, the limit I'm taking keeps blowing up to infinity. How should this problem be set up?
7
votes
4answers
683 views

Limit to infinity with natural logarithms

I found the following problem in my calculus book: Solve: $$\lim_{x\to \infty } \left(\frac{\ln (2 x)}{\ln (x)}\right)^{\ln (x)} $$ I tried to solve it using log rules and l'Hôpital's rule with no ...
3
votes
4answers
77 views

How to prove that the line perpendicular to the radius is the tangent in the calculus sense?

Let $P=(p_1,p_2)$ be a point on an semicircle and $r$ be the line perpendicular to the radius $\overline{OP}$, like the picture below. Euclid showed (Book III, Proposition 16) that $r$ does not ...
1
vote
4answers
122 views

What is the value of $\lim_{x\to 0}x^x$?

Evaluate $$\lim_{x\to 0}x^x$$ I tried by writing $x$ in terms of exponentials: $x=e^{\ln x}$ so $x^x=e^{x\ln x}$ $\lim_{x \to 0}x \ln x=\lim_{x \to 0}(\ln x +1) =-\infty$ Thus $\lim_{x\to ...
-5
votes
1answer
56 views

what is the limit of $(-2)^{1/(2n+1)}$ as $n\rightarrow\infty$? [on hold]

what is the limit of $(-2)^{1/(2n+1)}$ as $n\in\mathbb{Z}, n\rightarrow\infty$? and what is the limit of $(-2)^{2/(2n+1)}$ as $n\in\mathbb{Z}, n\rightarrow\infty$?
1
vote
3answers
85 views

limit of an integral question

Let $f : [0, \infty) \to \Bbb R$ be bounded and continuous. Prove that $\lim \limits _{h \to \infty} h \int \limits _0 ^\infty e ^{-hx} f(x) \, d x = f(0)$. Our intuition was to use l'Hospital's rule ...
-1
votes
1answer
44 views

Limit and summation. [on hold]

I find This limit and I want to get the value of it $$\displaystyle\lim_{n\to\infty} \displaystyle\sum_{i=1}^{n} \frac{7i^{3}+i^{2}+3i +1}{n^{4}+i+5} =? $$ I tried Riemann susummation t still can't ...
0
votes
0answers
55 views

Prove the existence of $\lim_\limits{x\to 3}{f(x)}$

Let $f:\mathbb{R}\mapsto \mathbb{R}$ be a function such that: $$f(x) \geq \frac{\left| x+1 \right| -2}{\left| x-3 \right|}, \forall x\in \mathbb{R-\{3\}}$$ Prove the existence and find the ...
0
votes
1answer
37 views

What does it mean to say $f(x) \sim g(x)$, i.e. $f(x)$ behaves like $g(x)$ when $x \to \infty$?

If $\lim_{x\to\infty}\frac{f(x)}{g(x)}=\infty$, then $f$ grows faster than $g$. Same if $\lim_{x\to\infty} \frac{g(x)}{f(x)} = 0$. Would $f$ behave like $g$ if $\lim_{x\to\infty}\frac{f(x)}{g(x)} = ...
4
votes
3answers
80 views

Challenging $\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}$ for $\epsilon=\frac{1}{2}$.

Consider the (incorrect) claim that $$\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}.$$ How might I find the largest $\delta$ such that I can challenge $\epsilon = 1/2$? Clearly ...
6
votes
5answers
111 views

Calculate the limit $\lim_{x\to 0} \left(\dfrac 1{x^2}-\cot^2x\right)$ [on hold]

The answer of the given limit is $2/3$, but I cannot reach it. I have tried to use the L'Hospital rule, but I couldn't drive it to the end. Please give a detailed solution! $$\lim_{x\to 0} ...
1
vote
3answers
62 views

Dealing with indeterminate forms of the $1^\infty $ kind

$$\lim\limits_{x→{\frac π{2}}^-}\left(\frac {2x}{\pi}\right)^ {\tan x}$$ and $$\lim\limits_{n→\infty} \left(1+ \frac {1}{n}\right)^n$$ could anyone provide some hints? how to start. (with ...
3
votes
1answer
89 views

When does $\exp\left(\sum_{i=1}^\infty a_i\right)=\prod_{i=1}^\infty \exp(a_i) ?$

Considering the complex logarithm, when do we have $$\exp\left(\sum_{i=1}^\infty a_i\right)=\prod_{i=1}^\infty \exp(a_i) ?$$ I originally wanted to try to prove it by showing $$\lim \prod^N ...
2
votes
2answers
81 views

Find $\lim_\limits{x\to 1}{\frac{f(x)-2\cdot x^2}{x-1}}$

Let $f: \mathbb{R}\mapsto \mathbb{R}$ be a function such that: $$\lim_\limits{x\to 1}{\frac{f^2(x)+f(x)-6}{x-1}}=5$$ If we know that $$\lim_\limits{x\to 1}{\frac{f(x)-2\cdot x^2}{x-1}}=a\in ...
0
votes
1answer
54 views

How to prove the limit exists for function of two variables?

Problem: Evaluate the indicated limit or explain why it does not exist: \begin{align*} \lim_{(x,y) \to (0,0)} \frac{x^2 y^2}{x^2 + y^4} \end{align*} The definition of limit my calculus textbook gives ...
1
vote
1answer
42 views

Integrals, intermediate value theorem question

f∈c[a,b] (f is continuous in [a,b]), prove: We tried to use the integral intermediate value theorem to try to prove it but we don't understand why the limit has to be the max and not any other value ...
2
votes
4answers
88 views

Prove that $\lim_\limits{x\to 0}{f(x)}=0$

Let $f:\mathbb{R}\mapsto \mathbb{R}$ be a function such that: $$2\cdot f(x)-\sin(f(x))=x, \forall x\in \mathbb{R}$$ Prove that $\lim_\limits{x\to 0}{f(x)}=0$. I think I need to use the sandwich ...
4
votes
1answer
44 views

Proving an equivalent definition of the $\lim_{x\to a}f(x)$ exists [duplicate]

Prove that the following statements are equivalent. (a) $\lim_{x\to a}f(x)$ exists (b) Given $\epsilon \gt 0$, there is a $\delta \gt 0$ such that if $0\lt |x-a| \lt \delta, 0\lt |y-a| \lt \delta$, ...
0
votes
0answers
35 views

Epsilon delta limit to show that [on hold]

show that $$\left|\frac{28}{3x+1}-4\right| = \left|\frac{12}{3x+1}\right| \cdot |x-2| $$ using $\epsilon$-$\delta$ definition of a limit. I have no idea where to start since the question is not ...
3
votes
5answers
91 views

Limit calculation: $\lim_{x\to 0}\frac{1}{x}\ln\left(\frac{e^x − 1}{x}\right)=$?

For some reason I'm having trouble calculating the limit of the following function : $$\lim_{x\to 0}\frac{1}{x}\ln\left(\frac{e^x − 1}{x}\right)$$ The function might, or might not converge. I've ...
4
votes
6answers
1k views

L'Hospital rule, exponental ratio

$$\lim_{x\to ∞} \frac {x^{1000000}} {e^x}$$ could anyone please provide some hits with what result I will end up? After all applyings of L'Hospital rule, I will get $\frac {n} {e^x}$, where $n$ is ...
5
votes
2answers
96 views

Substitution for limits [duplicate]

How does substitution for limits exactly work? I see often answers that use the substitution $t=\frac1x$, then changing $x\rightarrow\infty$ to $t\rightarrow0^+$. I have seen this question, this ...
4
votes
4answers
42 views

limit of $x \cot x$ as $x\to 0$.

I was asked to calculate $$\lim_{x \to 0}x\cot x $$ I did it as following (using L'Hôpital's rule): $$\lim_{x\to 0} x\cot x = \lim_{x\to 0} \frac{x \cos x}{\sin x} $$ We can now use L'Hospital's ...
0
votes
0answers
18 views

Finding limit and maximizer

Let $f(x):=x^\alpha - k \cdot (x+c)^\alpha$, defined for $x>0$, where $k,c>0$ and $0<\alpha<1$. Question: solve $\max_{x>0} f(x)$. Below are my thoughts: Calculate $f'(x) = \alpha ...
2
votes
3answers
85 views

$\lim_{x\to 0}\frac{e^x-1}{\sin x}$ equal to $\lim_{x\to 0}\frac{e^x-1}{x}$ because $x$ and $\sin x$ tend both to $0$ for ${x\to 0}$

I'm stuck in this limit: $$\lim_{x\to 0}\frac{x(e^x-1)}{\cos x-1}$$ I tried to solve it using special limits, so: $$\lim_{x\to 0}\frac{x(e^x-1)}{\cos x-1}=$$ $$=\lim_{x\to 0}(e^x-1)\frac{x(\cos ...
2
votes
1answer
34 views

Does convergence in probability preserve the weak inequality?

Suppose I have two sequences of random variables $\{x_n\}$ and $\{y_n\}$ such that $x_n\leq y_n$ and $\text{plim}\;x_n=L_x$ and $\text{plim}\;y_n=L_y$, can I say $L_x\leq L_y$ (almost surely)? Does ...
1
vote
0answers
34 views

Integral of a function which is everywhere discontinuous?

Yesterday, I tried to carry out a little thought experiment when it came to taking limits and have found that it has pushed my understanding of them to the breaking point. I tried considering the ...
0
votes
0answers
13 views

Basic limits as part of $\omega$-limit sets of dynamical systems

I am looking at a question and solution of an $\omega$-limit set of a set $I \in X$ in dynamical systems. I have the full solution but there is something I am a little stuck on. Here is the question: ...
1
vote
2answers
124 views

Find $\lim_{n \to \infty} n[(1+\frac{1}{n})^n - e]$

$\lim_{n \to \infty} n[(1+\frac{1}{n})^n - e]$ I let, $x = \frac{1}{n}$, then as $\lim_{x \to 0} \frac{1}{x}[(1+x)^\frac{1}{x} - e] = \infty$ L'hopital's: $\lim_{x \to 0} ...
-3
votes
2answers
37 views

Crazy and difficult Limits and integration

This limit take from me much time to solve and finally I can't. So please help me to solve.. Find $L$ $ L =\displaystyle\lim_{x\to \infty} \frac{\displaystyle\int_{1}^{x} t^{t-1} ( t + tln (t) +1 ) ...
-1
votes
1answer
22 views

Differentiation Derivation - Limits Question [duplicate]

Hi, I encountered these perplexing questions in my study of the derivation of trigonometric differentiation. Could someone help?
3
votes
2answers
221 views

Solving Limits with L'Hospital's Rules

I am having difficulties solving this limit. I was given the question and equation: Try using L’Hospital’s Rules to evaluate the follwing limit: $$\lim\limits_{u \to \infty } \frac{u}{\sqrt{u^2 ...
2
votes
2answers
83 views

Is $\lim S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}<\infty $ for $ n \to \infty$ and $m$ large?

Let $m$ be a fixed positive integer ($m>1)$ and let $$S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}$$ be a partial sum of real series. My question here is : Is $\lim S_{n,m} <\infty $ as $ n ...
0
votes
1answer
31 views

Can a limit of multivariable function can be taken componentwise?

Is there a theorem saying that $\lim_{(x,y)\rightarrow(p,q)} f(x,y)= \lim_{x\rightarrow p}(\lim_{y\rightarrow q} f(x,y))$? If so, could someone link me to a proof of it or give me a proof? Edit: So ...
3
votes
3answers
68 views

Using the definition of derivative to find $\tan^2x$

The instructions: Use the definition of derivative to find $f'(x)$ if $f(x)=\tan^2(x)$. I've been working on this problem, trying every way I can think of. At first I tried this method: $$\lim_{h\to ...
2
votes
2answers
58 views

Spivak's 'Calculus', 5-21(b): Is there an easier/shorter way?

Some personal background: I'll be going into my second year as a maths undergraduate in September of this year, and I'm currently working my way through Spivak's Calculus. While $\epsilon$-$\delta$ ...
0
votes
0answers
41 views

Limit approach to infinity [closed]

During my studying to limits I find this limit but I want to know How we can know that this limit is exist??? $$\lim_{x\to \infty} \sqrt{1-\cos\frac{1}{x} \sqrt{1-\cos\frac{1}{x} ...
0
votes
1answer
29 views

limit of sequences with number of pairs [closed]

Let $a_n$ be the number of pairs $(s,t)$ where $s$, $t$ are integers such that $s^2 + t^2 \leq n^2$ and $|s|+|t|\geq n$. Evaluate $\lim_{n \to \infty} \frac{a}{n^2}$.
2
votes
3answers
65 views

Find $\lim_\limits{x\to x_0}{\frac{x f(x_0)-x_0 f(x)}{x-x_0}}$

Let $f:\mathbb{R}\mapsto \mathbb{R}$ be a function such that: $$\lim_\limits{h\to 0}{\frac{f(x_0+h)-f(x_0)}{h}}=2016$$ Prove the existence of the following limit: $$\lim_\limits{x\to ...