Questions on the evaluation of limits.

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4
votes
1answer
54 views

Is this a valid proof of $\lim _{n\rightarrow \infty }(1+\frac{z}{n})^n=e^z$?

Define the function $g_n(z)=(1+\frac{z}{n})^n$ for $\:n\in \mathbb{R^+}$. Then $\frac{d}{dz}g_n(z)=n(1+\frac{z}{n})^{n-1}\cdot\frac{1}{n}=(1+\frac{z}{n})^{n-1}$ Define $g_{\infty}(z)=\lim ...
6
votes
3answers
438 views

How do you calculate this sum?

How to find the value of $S(\infty)$, where $S(n)$ is the following $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$ Wolfram alpha is unable to calculate it. This is a question from a ...
2
votes
2answers
61 views

Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$

Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$ I'm trying to solve it by using the squeeze theorem but I'm stuck. I'm looking for a function $g(x)$ such that $g(x) \leq ...
1
vote
1answer
30 views

Problems understanding definition of limit superior.

I'm aware that there are multiple questions on this topic and I have read most of them, but I still don't really understand the definition and would really appreciate some quick help with this, ...
0
votes
2answers
51 views

Problem on a limit. Can anyone solve it?

$$\lim_{x\rightarrow 0}\frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}=\,?$$ by directly substituting $x=0$ i got $\infty$ by using L-H's rule, i got $-1/8$ the given options are $a)\frac{24e}{11}$ ...
2
votes
1answer
38 views

Question on continuity with [x]

given the function $f(x)=\frac{2[x]}{3x-[x]}$ the question is to find continuity of the function at $x=1$ and $x=\frac{-1}{2}$ note: [x] denotes the largest integer which is less than or equal to x. ...
1
vote
2answers
50 views

Limit of a recursively defined sequence

Let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence of real numbers such that: $$\forall n\in\mathbb{N},\quad a_{n+1}=a_n + e^{-a_n}.$$ Prove that: $$\lim_{n\to+\infty}\frac{a_n}{\log n}=1.$$
5
votes
4answers
172 views

Another method for limit of [e-(1+x)^(1/x)]/x as x approaches zero

I have solved this limit: $\lim_{x \rightarrow 0} \frac{e-(1+x)^{\frac{1}{x}}}{x}$ using Hopital rule and series expansion. Do you have other method for solving it? Thanks.
5
votes
3answers
569 views

Why does this infinite series equal one?

Why does $$\sum_{k=1}^\infty \binom{2k}{k} \frac{1}{4^k(k+1)}=1$$ Is there an intuitive method by which to derive this equality?
1
vote
2answers
54 views

Calculus - Limit calculate help

I'm having a problem to solve this limit. $$\lim_{x \to \pi/4} \frac{\tan x-1}{\sin x-\cos x}$$ $\lim_{x \to \pi/4} \frac{\tan x-1}{\sin x-\cos x}$ = $\lim_{x \to \pi/4} \frac{\frac{\sin x}{\cos ...
4
votes
1answer
43 views

write this limit based on delta function

If it’s possible, I want to write this limit based on delta function $$ \lim_{t\rightarrow 0}\frac{e^{-u/t}}{t^2},\qquad u>0 $$ would you mind helping me?
2
votes
3answers
216 views

Find the limit of using L'Hopital's Rule

Could anyone explain to me how to calculate the limit $$\lim_{x \to 0} \frac{1}{\sqrt{x^3}} - \frac1{\sin x}$$
1
vote
4answers
158 views

Find limit of fraction without derivatives

How to find limit of this? \begin{equation} \lim_{h \rightarrow 0} \frac {\sqrt[3]{x+h} - \sqrt[3]{x}} {h}. \end{equation} I see here the derivative definition, so I understand in "derivative mind" ...
0
votes
0answers
22 views

Show that $ \prod_{k=1}^n [1 + p_{nk}(e^{it} - 1)] \rightarrow e^{\lambda(e^{it} - 1)}, n \rightarrow \infty $

Suppose that $0\leq p_{nk} \leq 1, 1 \leq k \leq n$, $\max_{1 \leq k \leq n} p_{nk} \rightarrow 0, n \rightarrow \infty$ and $\sum_{k=1}^n p_{nk} \rightarrow \lambda$. Show that $$ \prod_{k=1}^n ...
0
votes
1answer
29 views

how to find uniform continuity

I have some questions on continuity. What is the difference between continuous and uniformly continuous function? Please explain with this question. Find $f(x)=x^2$ is uniformly continous on ...
3
votes
2answers
40 views

Find two functions given properties of their limits

I've been stuck on a seemingly simple problem regarding limit properties of two functions: Find two functions, f(x) and g(x), given the following properties: $$\lim\limits_{x \to 7} f(x)=0$$ ...
1
vote
1answer
31 views

Logarithmic integral and natural numbers.

Prove these two relations: $$\text{li}(k+1)+k-\log (k)-\gamma = \int_0^k \left(\int_1^2 \frac{(s+1)^{n-1}+s-1}{s} \, dn\right) \, ds$$ $$n = \lim_{s\to 0} \, \frac{(s+1)^{n-1}+s-1}{s}$$ ...
1
vote
2answers
56 views

Cannot find the limit because the denominator is $0$

I need to solve one limes equation but I cannot find the way. Can you help me and explain me the way, please. The equation is this: $$\lim_{x\to -2} ...
1
vote
0answers
25 views

Complex numbers product and ratio, prove this relation.

Define a table $T$ as follows: $$\text{If}\; n\geq k \; \text{then} \; T(n,k) = (2+3 i) \sum _{i=1}^{n-1} T(n-i,k-1)+(5+7 i) \sum _{i=1}^{n-1} T(n-i,k) \; \text{else} \; T(n,k) = 0$$ Then take rows ...
0
votes
2answers
71 views

How to evaluate a limit with high school math?

So my textbook's explanation of the derivative of e is very sketchy. They used lots of approximations and plugging things into the calculator. Basically I want to know how you can work out as h ...
1
vote
0answers
57 views

How do I solve the limit of $\cfrac{r^n}{n!} \cfrac{(n-1)!}{(1-r)^n}$ as $n \rightarrow \infty$ when $0 < r < 1$?

I'm watching a coursera class where the professor is talking about using the radius of convergence $R$ to determine where the Taylor series expansion of a function equals the function. The function is ...
0
votes
1answer
39 views

Question about writing proofs for limit

I intuitively understand proof with limits, but I'm not sure on how to write a formal proof for this example. For each $n \in \mathbb{N}$, let $a_n$, $b_n$ be real numbers. Also, let $a_{\infty}$, ...
3
votes
2answers
52 views

Proof $e^n*n!$ is an asymptote of $(n+1)^n$

I would like to prove $\lim_{n\to \infty}e^nn!-(n+1)^n=0$. All I have really done is show $(n+1)^n=\sum_{i=0}^n\frac{n!}{(n+1)^i(i!)(n-i)!}$
1
vote
5answers
109 views

Limit of factorial function: $\lim\limits_{n\to\infty}\frac{n^n}{n!}.$ [duplicate]

I am studying for a test and I am given this problem: $$\lim_{n\to\infty}\frac{n^n}{n!}.$$ How do I go about solving this limit? Intuitively I see how the numerator is growing much faster, but how ...
0
votes
1answer
67 views

Quick question about $\epsilon -\delta$ proofs

There is one step in $\epsilon - \delta$ proofs that I hope somebody could bring clarity to for me. Say we wanted to show $\displaystyle \lim_{x \to 2} x^2 = 4 $. Somewhere along the proof we would ...
5
votes
3answers
117 views

Find the value of this infinitely nested radical (it appears to obtain multiple values)

Find the value of $$\sqrt{1-\sqrt{\frac{17}{16}-\sqrt{1-\sqrt{\frac{17}{16}-\cdots}}}}$$ This is not as simple as it looks for one reason - there are $2$ real solutions to the equation ...
4
votes
2answers
39 views

Simple series divergence problem

I've got a problem here: $$\sum_{n=1}^{\infty} \frac{5^n}{n(3^{n+1})}$$ I've used the ratio test and essentially did this: $$\sum_{n=1}^{\infty} \left( \frac{5^{n + 1}}{n (3^{n+1+1})} / ...
3
votes
2answers
71 views

Evaluate $\lim_{x \to 1} \frac{\sqrt[3]{x} -1}{\sqrt{x} -1}$

Evaluate $\lim_{x \to 1} \frac{\sqrt[3]{x} -1}{\sqrt{x} -1}$ I want to solve this limit by employing the strategy of introducing a new variable $t$ in such a way as to make the problem simpler. I've ...
0
votes
2answers
31 views

Valid way of evaluating limits?

Calculate the following limits $$\lim_{x \to 0} \frac{e^{\sin x} - \sin^2x -1}{x},\,\,\,\,\,\,\,\, \lim_{x\to0} \frac{\sin x \cos x - x}{x^2 e^x}.$$ I've evaluated these using the asymptotic ...
1
vote
2answers
98 views

Epsilon-Delta continuity definition for straight lines parallel to axes

I am taking a course on real analysis online and I encountered the $\epsilon-\delta$ definition for a function to be continuous. But I wonder if I can apply it to functions which are straight lines ...
-1
votes
0answers
50 views

$ max \{x : 0 \leq x < 1\} = ? $

As per the title, what is the maximum value: $$ \max \, \{x : x \in \mathbb R, 0 \leq x < 1\} = ? $$ This question came to me when considering the supremum metric applied to the set of functions ...
-1
votes
2answers
91 views

Evaluate $\lim_{x\to 2}\frac{x^3 - 8}{x-2}$.

$$ \lim_{x \to 2}\frac{x^3 - 8}{x-2}$$ I solved it my answer is 8. But the solution says it is 12. How?
3
votes
2answers
74 views

Let $a_{n+1}=cos(a_n)$ for $n\ge0$ and $a_0 \in [-\pi/2,\pi/2]$ Find $\lim_{n \to \infty}a_n$ if it exists.

Let $a_{n+1}=\cos(a_n)$ for $n\ge0$ and $a_0 \in [0,\pi/2]$ Find $\lim_{n \to \infty}a_n$ if it exists. I drew some sketches and it does seem like the limit exists, it's probably $x$ such that ...
0
votes
3answers
30 views

I do not quite understand this difference in limits

According it my study material: $\lim_{x\to 0^-}\frac {x}{|x|}= -1$ and $\lim_{x\to0^-} \frac {1}{|x|}= \infty$ Why does $\lim_{x\to0^-} \frac {1}{|x|}\ne -\infty$ as 1 still devided by a negative ...
1
vote
1answer
42 views

Evaluating $\lim_{h\rightarrow 0}\frac{2^{8\cos(h)}}{8h}\left [ \sin^{8}(\pi/6+h))-\sin^{8}(\pi/6) \right ]$

$$\lim_{h\rightarrow 0}\frac{2^{8\cos(h)}}{8h}\left [ \sin^{8}(\pi/6+h))-\sin^{8}(\pi/6) \right ]$$ My Attempt: For $\lim_{h\rightarrow 0}\frac{\sin^{8}(\pi/6+h)-\sin^{8}(\pi/6)}{h}=f'(x)=8\cdot ...
2
votes
1answer
48 views

approximation of $\log(1+z)=z$ as $z\to 0$

This is new to me and I have not done any asymptotic approximation. I don't understand how they get that $\frac{n}{N}$ stays close to $\frac{2}{3}$ as N goes to infinity. Also how do they do get ...
1
vote
4answers
91 views

Evaluate the limit: $\lim_{x\to \infty}$

Evaluate the limit: $$\lim_{x\to\infty} \frac{(2x^2 +1)^2}{(x-1)^2(x^2+x)}$$ The answer is 4 and I don't understand why, but why can't I just do something like:$$\frac{(\infty)}{(\infty)(\infty)} = ...
0
votes
1answer
45 views

Question about proofs with limits

I intuitively understand proof with limits, but I'm not sure on how to write a formal proof for this example. For each n $\in$ $\mathbb{N}$, let $a_{n}$ be a real number. Also, let $a_{\infty}$ be a ...
-3
votes
1answer
69 views

How to calculate these limits using L.Hopital Rule [on hold]

How to evaluate the limits, using L'Hospital rule? $$\lim_{x \rightarrow 0} \frac{x-\sin x}{x^2}$$ $$\lim_{x \to 0^+}\frac{\ln(x^2 + 2x)}{\ln x} $$
2
votes
3answers
49 views

How to evaluate $\lim_{x \to \infty}\left(1 + \frac{2}{x}\right)^{3x}$ using L'Hôpital's rule?

I'm stuck on how to evaluate the following using L'Hôpital's rule: $$\lim_{x \to \infty}\left(1 + \frac{2}{x}\right)^{3x}$$ This is a problem that I encountered on Khan Academy and I attempted to ...
1
vote
3answers
65 views

For What Values Of $x$ Is $f$ Continuous

For what values of $x\in\mathbb{R}$ is $f$ continuous? $f(x) = \left\{ \begin{array}{lr} 0 & \text{if}\, x \in \Bbb Q\\ 1 & \text{if}\, x \notin \Bbb Q \end{array} ...
1
vote
3answers
57 views

Proving a limit exists - solving for epsilon with absolute values

I have the equation that I want to prove the limit goes to 1: $$\lim_{n \to \infty} \frac {(n+8)(n+1)}{n(n-10)} = 1$$ Using definition of limit, I get this equation: $$ \left | \frac ...
0
votes
1answer
50 views

The definitions of limit infimum and limit supremum

I have begun reading Rosenthal's "A First Look to Rigorous Probability Theory" and in order to reinforce my calculus background I am studying through the appendix section. Here he defined the limit ...
1
vote
3answers
43 views

Finding Horizontal/Oblique Asymptote of $y=\frac{\sqrt{x}+1}{\sqrt{x}-1}$

Is it correct to simply subsitute $\sqrt{x}$ with $x$ when finding horizontal or oblique asymptotes? The method works but I am not sure if it is formally sound enough to pass muster in an examination. ...
2
votes
1answer
95 views

Evaluate the limit $\lim \limits_{x \to \infty} \frac{1}{x(x+1)}$ [closed]

How can I evaluate the limit $$\lim_{x \to \infty} \frac{1}{x(x+1)}$$
1
vote
5answers
65 views

How to evaluate the limit $\lim_{x \to \infty} \frac{2^x+1}{2^{x+1}}$

How to evaluate the limit as it approaches infinity $$\lim_{x \to \infty} \frac{2^x+1}{2^{x+1}}$$
2
votes
2answers
80 views

Exactly How Does This Proof Mean That The Cosine Function Is Continuous

The question is: Prove that cosine is a continuous function. To give some context in what way this must be answered, this question is from a sub-chapter called Continuity from a chapter introducing ...
2
votes
2answers
61 views

Compute limit of a function

Compute: $$\lim_{x \rightarrow 0^+} \frac{\arctan(e^x+\arctan x)-\arctan(e^{\sin x}+\arctan(\sin x))}{x^3}$$ WolframAlpha tells me it's 1/6. Any nice idea how to rewrite that expression? Thanks!
3
votes
2answers
25 views

Basic limit question to understand the methods

I have a very basic question about proving limits with the epsilon-delta method. So i want to prove $\lim _{x\to 0}\left(\frac{1}{1-2x}\right)\:=\:1$ . first, i write it like that: ...
1
vote
5answers
62 views

How to get the following limit into indeterminate form?

I am struggling to get the following limit into its indeterminate form so that i can apply the l'Hopitals rule: $$\lim_{x\to 0^+}(\sin x)^x$$ A solution would be greatly appreciated, been struggling ...