Questions on the evaluation and properties of limits.

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3
votes
1answer
20 views

Prove with Cauchy's limit definition ($\epsilon, \delta$) that $\lim_{x \rightarrow 0} \frac{x^2-8}{x-8}=1$

Prove with Cauchy's limit definition ($\epsilon, \delta$) that $$\lim_{x \rightarrow 0} \frac{x^2-8}{x-8}=1$$ Got really troubled with the proper technique of solving this. Any assistance will be ...
0
votes
1answer
12 views

Special case of limsup inequality

If you are given two sequences $a_{n}$ and $b_{n}$ such that $a_{n}b_{n} \geq 0$ and the limits $\lim\limits_{n \rightarrow \infty}a_{n} = a$ and $\lim\limits_{b \rightarrow \infty}b_{n} = b$ then can ...
-2
votes
4answers
42 views

Prove that lim of x/(x+1) = 1 as x approaches infinity

I want to prove that $$\lim\limits_{x\to \infty} \frac{x}{x+1}=1$$ I know that I need to show that: $$\left|\frac{1}{x+1}\right| \lt \epsilon$$ But I'm not sure how to manipulate it. Any help or hint ...
7
votes
1answer
44 views

$a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$

Suppose that $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$. I can find $\lim_{n \rightarrow \infty}a_n=0$. But I have no idea to find $\lim_{n \rightarrow ...
5
votes
1answer
45 views

How do I follow part of this simple proof that $\lim\limits_{n\to \infty} (a_n b_n) = a b$?

I'm working through some analysis textbooks on my own, so I don't want the full answer. I'm only looking for a hint on this problem. In Rosenlicht's Introduction to Analysis, he proves, in a few ...
0
votes
1answer
36 views

Prove that $\lim_{x \to 0}\frac{x^2+x^3}{\sin^3x}$ doesn't exist.

Prove that $$\lim \limits_{x \to 0}\frac{x^2+x^3}{\sin^3x}$$ doesn't exist, also ( in convergence to infinity), in three ways: 1) Using Cauchy's definition (by $\delta, \epsilon$). 2) According to ...
-1
votes
0answers
18 views

I need to find a rational numbers series that converging to irrational number [duplicate]

I found a series that is an+1=(an^2 + 2)/2an yet I'm not sure. can someone give me a more umm solid example? thanks.
0
votes
0answers
29 views

Evaluate $\lim_{n\to1^+}\left({\zeta(n)-\dfrac{1}{n-1}}\right)$

Let $$x=\lim_{n\to1^+}\left({\zeta(n)-\dfrac{1}{n-1}}\right)$$ where $\zeta$ is Riemann zeta function. What is the value of $x$? At $n\to1^+$, $\zeta(n)\to\infty$ and $\dfrac{1}{n-1}\to\infty$, so ...
0
votes
1answer
50 views

Proof via epsilon delta.

How do I formally prove this by using epsilon and delta? $a\neq 0$, if $\lim \limits_{x \to 0}f(x)= L$, then $\lim \limits_{x \to \infty}f(\frac{a}{x})= L$
3
votes
9answers
159 views

Why doesn't $e=1$?

I'm sure that this is a very basic question, but it has been bothering me for a while: If $e=\lim\limits_{x\to \infty} (1+x^{-1})^x$, shouldn't $e=1$? If $x$ is tending towards infinity, why ...
9
votes
2answers
208 views

To evaluate limit of sequence

How do I evaluate the limit of the following sequence $$a_n = \left(\left( 1 + \frac1n \right) \left( 1 + \frac2n \right)\cdots\left( 1 + \frac nn \right) \right)^{1/n}$$ I tried to take log and ...
0
votes
2answers
32 views

Epsilon delta proof for a function's limit.

I tried to solve the follwing limit of function by delta and epsilon but I got stuck with finding the right "N" that for each $x<N$, $|f(x) - L|< \epsilon$, and the hard part was how to get x ...
5
votes
5answers
120 views

Prove that $c_n = \frac1n \left(\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}} \right)$ converges

I want to show that $c_n$ converges to a value $L$ where: $$c_n = \frac{\large \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}}}{n}$$ First, it's obvious that $c_n > 0$. I ...
3
votes
4answers
71 views

Calculate $\lim_{x \to 0} (e^x-1)/x$ without using L'Hôpital's rule

Any ideas on how to calculate the limit of $(e^x -1)/{x}$ as $x$ goes to zero without applying L'Hôpital's rule?
0
votes
1answer
46 views

Reasoning behind multiplying by conjugates

What is the reason behind multiplying by conjugates? I am currently studying single variable calculus and throughout the lessons from the text I'm using, the reasoning as to why one would multiply by ...
0
votes
2answers
63 views

Prove limit doesn't exist using $\delta$-$\varepsilon$

Prove in the $\delta$-$\varepsilon$ definition that the limit $$\lim_{x \to \infty} \frac{3}{2+\sin(x)}$$ does not exist. I know that $\sin x$ gets different values as $x$ approaches infinity but ...
0
votes
0answers
37 views

Limit as x goes to infinity of (50)(cosx)^3/(x^2+10)

How do I find the limit as x goes to infinity of $$\frac{50(\cos x)^3}{x^2+10}$$? I have tried dividing by $1/x$, so it becomes ($50(\cos x)^3)/(x)/(x +\frac{10}x)$. But I seem to be stuck from ...
1
vote
0answers
29 views

Slope Formula Approaches Value of Derivative at a Point

I came across this question while helping a friend study for an Analysis exam; Analysis is not exactly my forte, so maybe I'm missing something obvious, I don't know: Suppose ...
1
vote
1answer
27 views

Can differential calculus (limits, integrals, derivatives) be encoded in lambda calculus?

I am wondering, if the Church-Turing thesis holds (all effectively calculable functions are computable by Turing machines/lambda calculus) and I can compute the limit of a function by hand, what is ...
10
votes
3answers
459 views

A limit with an intuitive and wrong answer

In my last question I asked about a limit used in my exploration of tangent circles and whatnot. I decided to come up with a more direct approach to my problem, and now I only have to evaluate the ...
5
votes
3answers
48 views

Limits using Maclaurins expansion for $\lim_{x\rightarrow 0}\frac{e^{x^2}-\ln(1+x^2)-1}{\cos2x+2x\sin x-1}$

$$\lim_{x\rightarrow 0}\frac{e^{x^2}-\ln(1+x^2)-1}{\cos2x+2x\sin x-1}$$ Using Maclaurin's expansion for the numerator gives: $$\left(1+x^2\cdots\right)-\left(x^2-\frac{x^4}{2}\cdots\right)-1$$ And ...
0
votes
1answer
19 views

Application of Bolzano's theorem in polynomials with the $a_n$ coefficient opposite.

Let $f(x)=x^2+βx+γ$ and $g(x)=-x^2+βx+γ$, where $γ \neq 0$. If $ρ_1$ is a real root of $f$ and $ρ_2$ a real root of $g$ with $ρ_1<ρ_2$, show that $f(x)+2g(x)=0$ has at least one real root in the ...
3
votes
3answers
70 views

Is $\lim_{n\rightarrow\infty}\sin\left(\pi\sqrt[3]{n^{3}+1}\right)$ exist?

I have that limit: $$\lim_{n\rightarrow\infty}\sin\left(\pi\sqrt[3]{n^{3}+1}\right)$$ I don't even know if it exists. If so, what its value ? Really don't have any idea..
3
votes
3answers
25 views

Finding a limit using change of variable- how come it works?

I'm a student just starting calculus in college, and my math skills are pretty stale. So... how come finding limits using change of variable works? For example: $$\lim_{x \to 1}\frac{x\cos(x-1) ...
2
votes
3answers
95 views

Show that $\lim_{n\rightarrow\infty}{\displaystyle\sum_{i=1}^{n}{\frac{F_n}{2^n}}}=2$

I need help to show that $\lim_{n\rightarrow\infty}{\displaystyle\sum_{i=1}^{n}{\frac{F_n}{2^n}}}=2$, where $F_n$ is the n-th number in the Fibonacci sequence. I know how to prove this by putting ...
-1
votes
2answers
43 views

Evaluating this limit

I have a calc 2 final soon and I've gotten the majority of the material down, except this one question on the review sheet. I just don't get how I would approach it. Evaluate the following Limit: ...
-2
votes
2answers
92 views

Prove that $\lim_{x\to\frac{2}{\pi}}\big\lfloor\sin\frac{1}{x}\big\rfloor=0$ [on hold]

Prove that $$\lim_{\large x\to \frac{2}{\pi}} \left\lfloor\sin\left(\frac{1}{x}\right)\right\rfloor=0$$ using the $\varepsilon$-$\delta$ definition of limits. Note that $\lfloor 0.1\rfloor = 0,\; ...
3
votes
2answers
85 views

Evaluate $\lim_{x\rightarrow 0^{+}}(3^{x}-2^{x})^{{1}/{x}} $

Evaluate $$\lim_{x\rightarrow 0^{+}}(3^{x}-2^{x})^{{1}/{x}} $$ I tried to use $\ln$ and $e$ with no success.
2
votes
1answer
32 views

Simple computation question about the limit of a function including little oh

Consider a sequence $$c_n:= t + o(t/n)n$$ where $o(\cdot)$ denotes little-oh I want to compute $\lim_{n\to \infty} c_n =?$ I guessing the result should be $\lim_{n\to \infty} c_n = t$ but not sure. ...
0
votes
3answers
23 views

Limit and L'Hopitals

I'm having trouble with this problem. $\lim{n \to \infty} (1+\frac 3n)^n$ My professor said to use a proof to figure out that the limit of the ln of the function is 3, but I can't figure out how to ...
2
votes
1answer
38 views

using power expansion to find limit

I am preparing for my final exam, and stuck on this question. Using power series expansion, evaluate $$\lim_{x\to 0} \frac{x\cos(x) -\sin(x)}{x^2-x\ln(1+x)}$$ I have no idea how to proceed. ...
2
votes
1answer
17 views

recursive sub-sequences of sequence , one is increasing and one is decreasing to same limit -> the sequence converge?

Let $b_1=\:0$, $b_{n+1}\:=\:\frac{1}{1+b_n}$. I need to show that $\left(b_n\right)_{n\:=1}^{\infty }$ converge. I thought about dived $b_n$ to 2 sub_sequence : $b_{2n}$, $b_{2n+1}$. (i thought ...
2
votes
1answer
80 views

Prove $\lim_{n\to\infty}[(n^2+n)^{1/2}-n]= 1/2$

So I know that this for all epsilon>0, there exists an N such that for all n>N dp$ (p_n,\frac{1}{2})<epsilon $. But in this particular example I'm having difficulty finding the $N$ for which this ...
0
votes
1answer
14 views

2 Sub-limits of sequence converge epsilon proof

consider $\left(a_n\right)_{n=0}^{\infty }$, $L_1,L_2\:\in \mathbb{R}$. $\lim _{k\to \infty }\left(a_{2k}\right)\:=\:L_1$, $\lim _{k\to \infty }\left(a_{2k-1}\right)\:=\:L_2$ . How to prove using ...
0
votes
0answers
31 views

Proving a limit of a trigonometric function

I need to prove the limit of this using the $\epsilon - \delta $ way but I don't know how to find $\delta$ when I'm given a trigonometric function I know only how to do it with polynomial functions
3
votes
2answers
49 views

Proving that for all complex $z$, $\lim_{x\to0}\frac{1-\cos^{z}x}{x^2}=\frac{z}{2}.$

What do I need to study beforehand in order to prove it (not necessarily in only one way)? I found this sperimentally, at the moment we're beginning derivatives at school. By induction, I succeeded in ...
0
votes
1answer
24 views

Please help me evaluate this product involving logarithms.

Please help me evaluate this product: $$\prod _{n=0}^{\infty } -\frac{\log \left(-\frac{1}{6 n+2}\right) \log \left(-\frac{2}{6 n+3}\right) \log \left(-\frac{1}{6 n+4}\right) \log \left(\frac{1}{6 ...
3
votes
1answer
52 views

Problem 3.14(e) in Baby Rudin

If $\{s_n\}$ be a sequence of complex numbers, define its arithmetic mean $\sigma_n$ by $$\sigma_n \colon= \frac{s_0 + s_1 \cdots + s_n}{n+1} \, \, (n = 0, 1, 2, \ldots). $$ Put $a_n = s_n - s_{n-1}$ ...
-6
votes
1answer
44 views

Is Infinity the limit? [on hold]

Few weeks ago I heard the phrase "Infinity is not the limit!" in a movie. Today I thought about it form the math side. Why the infinity is not the limit? How can I prove it? Thanks.
0
votes
2answers
65 views

How to prove that $\lim_{x \to \infty} x = \infty$

Please refrain from using logic symbols, as I do not understand those. So, this is the question: $$\lim_{x \to \infty} x = \infty$$ Proving this using the actual formal definition of a limit. So ...
4
votes
1answer
49 views

Evaluate this limit in terms of f

I want to evaluate the following limit: $$\lim_{d\to x} \dfrac{\dfrac{2x}{f'(x)}+f(x)-f(d)-\dfrac{x^2-d^2}{f(x)-f(d)}}{2\left(\dfrac{d-x}{f(x)-f(d)}+\dfrac{1}{f'(x)}\right)}$$ I tried L'hopital's ...
0
votes
1answer
19 views

Definition of limit points and isolated points

Definition of isolated points: If the points $p \in E \subset X$ is not a limit point, then $p$ is called isolated point of $E$. My question is.. Then all the points in open set is called interior ...
0
votes
3answers
56 views

Explain sandwich theorem

I was reading my math book trying to understand "limits and derivatives". I understood almost everything till this. Below is the statement from my book. Can anyone please explain this to me. If ...
2
votes
2answers
59 views

Limit of an integral

I'm not sure how to approach (no pun intended) the following limit: $$\lim_{x \to 0^{+}} \sqrt{|\sin x - \tan x | } \int_{\cos x}^{1+ \sin x} e^y \, \, \mathrm{d}y$$ I know that the indefinite ...
2
votes
4answers
37 views

How to find $\lim_{t \rightarrow 0} (ax^t + by^t)^{\frac{1}{t}}$ with different methods?

Suppose $a,b,x,y$ are all positive and $a+b=1$. Compute $\lim_{t \rightarrow 0} (ax^t + by^t)^{\frac{1}{t}}$. I tried to put this into a form where L'Hôpital could be useful, but I was unable to do ...
1
vote
2answers
108 views

$\displaystyle\lim_{n\to\infty}a^{n}\left(\frac{n+1}{n}\right)^{n^{2}}$ converges for $a$ in what range?

$\displaystyle\lim_{n\to\infty}a^{n}\left(\frac{n+1}{n}\right)^{n^{2}}$ converges for $a$ in what range? I tried $\displaystyle\lim_{n\to\infty}\ln ...
1
vote
2answers
45 views

Clarification about notation for one-sided limits

Is $\lim_{x \to 3-0} f(x)$ the same as $\lim_{x \to 3^-} f(x)$, and is $\lim_{x \to 3+0} f(x)$ the same as $\lim_{x \to 3^+} f(x)$? Could anyone clarify this for me please? Thanks
2
votes
1answer
24 views

Is it possible to find the limit of this?

After recently watching a Numberphile video about a square problem I started thinking about what would happen to the sum of all angles if you had n amount of squares. After a bit of testing, I ...
1
vote
2answers
61 views

Can all limits be solved without L'Hopital?

Is there any type of limit that requires L'Hopital's rule to solve or can all limits be solved without using it? If all limits can be solved without L'hopitals, is there some sort of proof or ...
1
vote
1answer
25 views

How to find $\lim\limits_{z \to z_0} \frac{{\overline z}^2-{\overline z_0}^2}{z-z_0}$

Fairly simple question, I want to find this limit $$\lim_{z \to z_0} \frac{{\overline z}^2-{\overline {z_0}}^2}{z-z_0}$$ The original question was to find the region at which the function ...