Questions on the evaluation of limits.

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0
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2answers
66 views

How do I solve $\lim$ as $x$ goes to infinity of $(\frac{1}{x})^{\frac{1}{x}}$ without appealing to L'Hôpital?

How do I solve $\lim$ as $x$ goes to infinity of $(\frac{1}{x})^{\frac{1}{x}}$ without appealing to L'Hôpital? Note: If I take natural logs of both sides, I eventually must invoke L'Hôpital. The ...
0
votes
0answers
19 views

Calculus Secant Line

The point P(4, 2) lies on the curve y=√x . If Q is the point (x,√x) , use your calculator to find the slope of secant line PQ (rounded to six decimal places) for the value of x=3.999. Select one: a. m ...
10
votes
7answers
678 views

There is no smallest infinity in calculus?

Somewhat of a basic question, but I tried mixing set theory and calculus and the result is a giant mess. From set theory (assume ZFC) we know there is a smallest infinite cardinal, $\aleph_0$, and ...
0
votes
3answers
24 views

Limit, Greatest Integer function?

Q. Find $\lim _{x\to 0}\left(1-x+\left[x-1\right]+\left[1-x\right]\right)$ where $\left[y\right]$ denotes the greatest integer function not exceeding 'y'.
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1answer
19 views

How to find the limit with limit on variable in power?

I was reading the paper Quantum Computational Complexity in the Presence of Closed Timelike Curves and I am unable to prove a limit in it: $$\lim_{x \to \infty}(1-\frac{s}{2^{n-1}})^{x}=0 \;\;\;where ...
0
votes
0answers
14 views

Limit value of ceiling function in a trigonometric function.

What is the Left Hand Limit(LHL) and Right Hand Limit(RHL) of the $\lim_{x->0}({\sin[x]}/{[x]})$? where $[x]$ is the greatest integer function(step function) I got LHL as $\sin(1)$. Can anybody ...
-2
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1answer
44 views

Find without L'Hopital $\lim_{x\rightarrow 0^+} \frac {\tan x - x}{ x -\sin x}$ [on hold]

Find without L'Hopital $$\lim_{x\rightarrow 0^+} \frac {\tan x - x}{ x -\sin x}$$
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2answers
49 views

Limit h to 0 question… Is this right?

$$ \lim_{h\to0}\frac{\sqrt{a+h}-\sqrt{a}}{h} $$ I can cancel the top out by multiplying by $$ \frac{\sqrt{a+h}+\sqrt{a}}{\sqrt{a+h}+\sqrt{a}} $$ and get $$ =\frac{a+h-a}{h(\sqrt{a+h}-\sqrt{a})} $$ ...
1
vote
0answers
21 views

homotopy group of the limit space

Let $V_k(\mathbb{R}^{n+k})$ be Stiefel manifold. Using $\pi_i(V_k(\mathbb{R}^{n+k}))=0$ for all $i\leq n-1$, how to obtain $\pi_i( V_k(\mathbb{R}^{\infty}))=0$ for all $i\in \mathbb{N}$? Can I just ...
2
votes
1answer
25 views

How to solve $\lim _{k\rightarrow 1}\dfrac {1+\ln k}{\left| \ln \left( \ln k\right) \right| }$

How to solve $\lim _{k\rightarrow 1}\dfrac {1+\ln k}{\left| \ln \left( \ln k\right) \right| }$ I stucked at the denominator.
4
votes
3answers
108 views

Limit $\lim_{{x\rightarrow 0}}{\left(\frac{\frac{1}{\sqrt{1+x}}-1}{x}\right)}$

I am a high school student in Calculus, and we are finishing learning basic limits. I am reviewing for a big test tomorrow, and I could do all of the problems correctly except this one. I have no ...
2
votes
1answer
48 views

Proving a limit with epsilon delta definition

I am in honors Calculus I and my teacher is really stressing this limit proof. I understand the examples she goes over in class but she gave us a problem for home work and i just dont know how to ...
3
votes
1answer
50 views

Limit of Lambert $W$ Product Log is the Natural Log?

In solving this equation $\large y=x^ne^x$ I get the result that $$n \cdot W\left( \frac{y^{1/n}}{n}\right)=x $$ So now it is apparent to me that when $n=0$ you would simply get $\ln(y)=x$ by ...
1
vote
1answer
32 views

Multivar limit $\frac{6x-2y}{9x^2-y^2}$ by approach

I'm resolving the limit of $\frac{6x-2y}{9x^2-y^2}$ when $(x,y)\to(1,3)$ We didn't study any special theorem, we did only approach. I tried first the changes $y=mx$ and $y=x^2$. In $f(x,mx)$ ...
2
votes
0answers
36 views

Question on limits and determining the constant

I am told to determine the value of constant $C$ for which the limit $\displaystyle \lim_{x \to0} g(x)$ exists. The function is $g(x) = e^{-x} + C$. Thanks in advance
0
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1answer
19 views

Multivar limit $\frac{(2x^2).y}{x^4+y^2}$

I'm resolving the limit of $\frac{(2x^2).y}{x^4+y^2}$ when $(x,y)\to(0,0)$ We didn't study any special theorem, we did only approach. I tried first the changes $y=x^2$ and $y=0$. In $f(x,x^2)$ ...
1
vote
2answers
35 views

Taking limits of powers containing 'x'?

If we have $lim_{x\rightarrow 0}\sqrt{x^2+sinx-tanx}$ we can write this as $\sqrt{lim_{x\rightarrow \:0}\left(x^2+sinx-tanx\right)}$ OR If we have $lim_{x\rightarrow ...
0
votes
1answer
30 views

Evaluating $\lim_{x \to 0^+} \left [ \frac \pi 2 + \tan x - \arctan \frac 1x \right]^{\frac 1 {\ln x}}$ without de L'Hopital?

Making use of the identity: $$\arctan x + \arctan \frac 1x = \left\{\begin{align} \pi/2, x \gt 0\\ -\pi/2, x \lt 0 \end{align}\right.$$ I evaluated the limit as follows: $$\begin{align} \lim_{x \to ...
0
votes
1answer
26 views

How to exactly use this theorem? Limits of sequences.

So the theorem I am having trouble understanding is If sequence $a_n$ converges and has the limit $L$, written $\lim_{n\rightarrow\infty}a_n=L$ if for every $\epsilon > 0$ there exists a ...
3
votes
1answer
32 views

Value of constants when a limit is finite?

If $\displaystyle \lim_{x\to0}\frac{a\cos x+bx\sin x-5}{x^4}$ is finite. Find the value of 'a' and 'b'. ...
-1
votes
3answers
25 views

Computing $\lim_{(x,y)\to(0,0)}(x^2+1)\cdot\frac{\sin y}{y}$

Hi I have a limit with two variables in front of me and the book says directly that it is equal with $1$ but for the life of me I dont understand why?? maybe the answer is stupid but I am excausted ...
3
votes
3answers
49 views

If $f$ satisfies $\forall x\in\Bbb{R},0\leq f'(x), f''(x)$ and if $\exists a\in\Bbb{R}$ such that $0<f'(a)$, Then $lim_{x\to\infty}f(x)=\infty$

I got this problem: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function that satisfy $\forall x\in\mathbb{R},0\leq f'(x)$ and $0\leq f''(x)$ Prove that if $\exists a\in\mathbb{R}$ ...
4
votes
5answers
78 views

Evaluating $\lim_{x \to 0} \frac{(1 + \sin x + \sin^2 x)^{1/x} - (1 + \sin x)^{1/x}}{x}$

I did this: $$\begin{align} \lim_{x \to 0} \frac{(1 + \sin x + \sin^2 x)^{1/x} - (1 + \sin x)^{1/x}}{x} &\sim \lim_{x \to 0} \frac{(1 + x + x^2)^{1/x} - (1 + x)^{1/x}}{x} = \\ &= \lim_{x \to ...
0
votes
0answers
19 views

Proof of the Existence of the Scalar Multiple of a Convergent Sequence

I'm curious whether it can be proven, given a convergent sequence {S$_n$}, that some multiple of the limit of this sequence exists. It seems like a pretty simple statement and I'm sure its possible ...
0
votes
3answers
63 views

Prove that $\lim \limits_{(x,y,z) \to (0,0,0)} \frac{{xyz}}{{x+y+z}}=0$

I have a strong feeling that the following limit is zero, can anybody help me prove it. $ \lim\limits_{(x,y,z) \to (0,0,0)} \frac{{xyz}}{{x+y+z}}$ Thanks!
-5
votes
2answers
35 views

Symbolically find delta in terms of epsilon [on hold]

I need proof the $$\lim_{x\rightarrow 0} (x^3+1)=1$$ symbolically using $\delta$ in terms of $\epsilon$. I think I understand the basic concept of this, however I am having a difficult time when I ...
0
votes
1answer
30 views

if $\lim _{n\to \infty }\left(b_n\right)=L$ than $\lim _{n\to \infty }\left(\frac{1}{b_n}\right)=\frac{1}{L}$?

it's begginer question i know but very important for me to understand: $b_n\:\ne 0,\:L\ne 0$ if $\lim _{n\to \infty }\left(b_n\right)=L$ than $\lim _{n\to \infty ...
2
votes
3answers
72 views

How do I compute $\displaystyle\lim _{x\to 0} \tfrac{e^x+\sin x -1}{\ln(1+x)}$?

I'm a Calculus I teacher's assistant. One of my students asked me how to compute this limit $$\lim _{x\rightarrow 0} \dfrac{e^x+\sin x -1}{\ln(1+x)}$$ I could not solve it. I need some hint. P.s: ...
0
votes
0answers
17 views

The limit as $x$ approaches $-\frac{3\pi}{4}$ of $\cos(4x-\cos(2x))$.

The limit as $x$ approaches $-\frac{3\pi}{4}$ of $\cos(4x-\cos(2x))$. For this problem, I can simply evaluate the function at $f(c)$, where $c=-\frac{3\pi}{4}$? In other words, $$\cos(-3\pi - ...
0
votes
2answers
30 views

$\lim_{h\to0} \frac{\sqrt[3]{x+h+3}-\sqrt[3]{x+3}}{h}\;?$

What is the fraction that have to multiply to calculate the limit $$\lim_{h\to0} \frac{\sqrt[3]{x+h+3}-\sqrt[3]{x+3}}{h}\;?$$
3
votes
3answers
191 views

Calculating the limit

I want to calculate the following limit: $\displaystyle\lim_{x\rightarrow +\infty} x \sin(x) + \cos(x) - x^2 $ I tried the following: $\displaystyle\lim_{x\rightarrow +\infty} x \sin(x) + \cos(x) - ...
2
votes
1answer
54 views

How to find limit of this integral???

I had this problem on my last exam,and I couldn't do it: $$\begin{align} \lim_{x \rightarrow \infty}\frac{\int_{0}^{x} e^{t^{2}}dt}{x^{5}\int_{0}^{x^{2}}\frac{e^{t}}{t^{2}} dt} \end{align}$$ ...
3
votes
2answers
29 views

Other ways to evaluate $\lim_{x \to 0} \frac 1x \left [ \sqrt[3]{\frac{1 - \sqrt{1 - x}}{\sqrt{1 + x} - 1}} - 1\right ]$?

Using the facts that: $$\begin{align} \sqrt{1 + x} &= 1 + x/2 - x^2/8 + \mathcal{o}(x^2)\\ \sqrt{1 - x} &= 1 - x/2 - x^2/8 + \mathcal{o}(x^2)\\ \sqrt[3]{1 + x} &= 1 + x/3 + \mathcal{o}(x) ...
-1
votes
4answers
99 views

Prove $\lim_{n\to\infty} 1.01^n = +\infty$ [on hold]

Prove $\lim_{n\to\infty} 1.01^n = +\infty$ It seems simple enough however it is giving me problems, any help?
2
votes
1answer
37 views

Proof of convergence of $a_{n+1} = \dfrac{a_n^2 + 1}{3}$ in $\mathbb{R}$ and finding its limit

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
2
votes
0answers
19 views

Calculating a limit with exponential terms

Let $a,b,c$ be non-negative numbers. What are the conditions on $a,b,c$ so that $$ \lim_{n\to\infty}\left[1-e^{-na}-e^{-nb}\right]^{\exp(nc)} = 1. $$ Using the fact that $$ \log(1+x)\geq\frac{x}{1+x} ...
-2
votes
3answers
47 views

Finding the values of $a$ and $b$ such that $f$ is continuous and differentiable at $x = 1$? [on hold]

The equation is $F(x) = \begin{cases} x^2 & \text{if } x \leq 1 \\ ax+b & \text{if } x>1 \end{cases}$ Differentiable at $x = 1$ I'm having a hard time understanding on how to ...
2
votes
3answers
44 views

$\lim _{(x,y)\to(0,0)}\frac{x(x-y)}{x^4+y^4}$ does not exists

Show that $$\lim_{(x,y)\to(0,0)} \frac{xy(x-y)}{x^4+y^4}$$ does not exists. I've tried the "traditional" paths, with $(x,x)$, $(0,x)$, $(0,-x)$, but I only get $0$ as answer. Any hint? Thanks!
0
votes
1answer
37 views

Prove that a function has limit everywhere.

I need to prove the following: Assume that $f: \mathbb{R} \to \mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ for all $x,y \in \mathbb{R}$. If $f$ has a limit at zero, prove that $f$ has a limit at every ...
0
votes
0answers
24 views

Finding a limit using Taylor's theorem

let's say that g(x,y) is $c^{n+1}$ and let's say that p(x,y) is it's n-th order Taylor polynomial. I am trying to prove that: $$\lim_{(x,y)\to (0,0)} \frac{g(x,y)-p(x,y)}{(\sqrt{x^2+y^2})^n}=0$$ I ...
0
votes
1answer
27 views

If $|P_{n+1}-q|\le c|P_n - q|$ for all $n$, where $c<1$, then $P_n\to q$

Given $|P_{n+1}-q|\le c|P_n - q|$ for all $n$, where $c<1$ show that the $$\lim_{n\to\infty} P_n=q.$$ Was told to complete this problem by iteration. I'm terrible with proofs and they don't make ...
1
vote
2answers
29 views

How to find the derivative of improper integral with variable upper limit?

I have the integral from $-\infty$ to $y^2$ of the function $(e^{-|x|})$ and I need to find the derivative of this. That is, $$\frac{d}{dy} \int_{-\infty}^{y^2} e^{-|x|}\,dx$$ Usually derivative ...
2
votes
2answers
32 views

How do I find $\lim\limits_{x \to 0} x\cot(6x) $ without using L'hopitals rule?

How do I find $\lim\limits_{x \to 0} x\cot(6x) $ without using L'hopitals rule? I'm a freshman in college in the 3rd week of a calculus 1 class. I know the $\lim\limits_{x \to 0} x\cot(6x) = ...
0
votes
3answers
20 views

How do i evaluate the lim using limit laws

I have been looking over the limit laws and watching videos but i cant find a similar problem. the question asks me to use limit laws to evaluate the limit $$\lim_{x\to 0}\frac{x}{\sqrt{1+3x}-1}$$ ...
0
votes
1answer
21 views

Using $\epsilon - \delta$ method prove that $\lim_{(x, y)\to (\frac{\pi}{2}, 0)}xy\sin(x + y) = 0$

Using $\epsilon - \delta$ method prove that, $$\lim_{(x, y)\to (\frac{\pi}{2}, 0)}xy\sin(x + y) = 0$$ Here's what I tried: $$|xy\sin(x + y)| \le |xy| \le \frac{|x|^2 + |y|^2}{2}$$ But I can't ...
2
votes
0answers
30 views

The closed form of $\lim_{x\to4/3}\frac{\partial}{\partial x}\left( _2F_1\left(\frac{1}{3},1;x;-1\right)\right)$

Do you think the following limit might have a closed form? Some hints, clues? $$\lim_{x\to4/3}\frac{\partial}{\partial x}\left( _2F_1\left(\frac{1}{3},1;x;-1\right)\right)$$
0
votes
1answer
36 views

The limit of a product is the product of the limits — need help understanding proof. [on hold]

The definition of limit says "Given any positive radius ε about L, there exists some positive radius $∂$ about $c$ such that for all $t$ within $∂$ units of $c$ (except $t = c$ itself) the values of ...
1
vote
2answers
59 views

Calculus limit epsilon delta

Prove using only the epsilon , delta - definition $\displaystyle\lim_{x\to 2}\dfrac{1}{x} = 0.5$ Given $\epsilon > 0 $, there exists a delta such that $ 0<|x-2|< \delta$ implies $|(1/x) ...
2
votes
2answers
58 views

Calculus Limits Problem

L'Hopital's Rule is not allowed. Question 1: $$\lim_{x\to -2} \frac{\sqrt{6+x}-2}{\sqrt{3+x}-1} = \ ?$$ I tried to cross multiply $\frac{\sqrt{6+x}-2}{\sqrt{3+x}-1}$with ...
2
votes
4answers
56 views

Proving $\lim_{n \rightarrow \infty} \frac{\sum_{r=1}^{n} r^a}{n^{a+1}}=\frac{1}{a+1}$

How do we prove that $$\lim_{n \rightarrow \infty} \dfrac{\displaystyle\sum_{r=1}^{n} r^a}{n^{a+1}}=\frac{1}{a+1}$$? This type of terms appear in problems on limits, but I am unable to prove this. ...