Tagged Questions
-1
votes
0answers
46 views
What is $\lim\limits_{|a| \to \infty} \int_0^1 (G(x) - a_0 - a_1x)^2\,dx$?
$$\lim_{|a| \to \infty} f(a) = \lim_{|a| \to \infty}\int_0^1 (G(x) - a_0 - a_1x)^2\,dx$$
$a= [a_0, a_1]$, as $|a| \to \infty$, we have $a_0^2 + a_1^2 \to \infty$.
When I expand I get $\int_0^1 (g^2 ...
0
votes
1answer
48 views
Compute limit of the sequence given by $x_1=1$, $x_{n+1}^2=\frac{x_n+3}{2}$
Let $\left(x_n\right)$ be a real sequence such that $x_1=1,x_{n+1}^2=\dfrac{x_n+3}{2},\forall n\geq 1$.Compute $\lim_{n\to+\infty} 3^n\sqrt{\dfrac{9}{4}-x_n^2}$
5
votes
5answers
112 views
Limit as $x$ approaches $1$ from the right of $\frac{1}{\ln x}-\frac{1}{x-1}$
$$
\lim_{x\rightarrow 1^+}\;\frac{1}{\ln x}-\frac{1}{x-1}
$$
So I would just like to know how to begin to solve this limit, or what topic does this problem fall under so that I can search for ...
4
votes
3answers
102 views
Limit. $\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$
I want to compute $$\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$$
Is it OK how I want to do?
...
4
votes
8answers
136 views
Limit of $\lim_{x \to 0}\left (x\cdot \sin\left(\dfrac{1}{x}\right)\right)$ is $0$ or $1$?
WolframAlpha says $\lim_{x \to 0} x\cdot \sin\left(\dfrac{1}{x}\right)=0$ but I've found it $1$ as below:
$$
\lim_{x \to 0} \left(x\cdot \sin\left(\dfrac{1}{x}\right)\right) = \lim_{x \to 0} ...
0
votes
1answer
61 views
Why do these trig functions “overpower” each other?
For example, $\sin(x)\cos(x)$ can be written as $\sin(2x)/2$, the limit as $x$ approaches $0$ of $\sin(x)\cos(x)$ is $0$, and the limit as x approaches $\pi/2$ is $0$. I don't see a reason why sine ...
1
vote
1answer
18 views
Decreasing from the horizontal asymptote
The function $f(x) = x^2/(x^2 - x -2)$ has the following graph. It has a horizontal asymptote $y=1$. For $x$ less than $-4$, the function is decreasing and its graph is under the asymptote. How is ...
3
votes
7answers
125 views
How to prove that $1/n!$ is less than $1/n^2$?
I want to prove
$$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test.
How to prove ...
2
votes
1answer
49 views
How to place a limit that it's inside the integral, outside.
I did this:
$$\int_{1}^t x^{-1}dx=\int_{1}^t\lim_{n\rightarrow -1}{x^n}dx =\lim_{n\rightarrow -1}\int_{1}^t{x^n}dx $$ just to have a way to approximate $\ln t$. $$\ln{t}=\lim_{h\rightarrow ...
1
vote
3answers
52 views
Limit as N goes to Infinity
Consider this limit: $$\lim_{n\rightarrow\infty} \left( 1+\frac{1}{n} \right) ^{n^2} = x$$
I thought the way to solve this for $x$ was to reduce it using the fact that as $n \rightarrow \infty$, ...
2
votes
2answers
110 views
Without calculating limit directly show that it is equal to zero
$$\lim_{n\rightarrow\infty}\left(\frac{n+1}{n}\right)^{n^2}\frac{1}{3^n}=0$$
I am not really sure what it means by "without calculating limit" and I don't really have ideas how to do it.
2
votes
3answers
107 views
How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$?
How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$ ?
i know that its "1", but why?
How can i calculate this?
Thank you very very much =)
0
votes
2answers
43 views
Find the limit of $2+\left(-\frac{2}{e}\right)^n$, as $n\to\infty$, if it exsists
I'm absolutely unsure about how to approach this. I've considered changing it to $-2=\left(-\frac{2}{e}\right)^n$ and then using the properties of lograrithms, but $\ln(-2)$ is undefined, as is ...
1
vote
2answers
36 views
Is it ever proper to say that the limit of a function equals infinity?
If I calculate a limit and get the value $\infty$, what is the proper way to communicate this? Can I say that the $\lim_{n\to\infty}a_n=\infty$ and therefore the sequence $\{a_n\}$ diverges, or do I ...
3
votes
3answers
27 views
Limit as n approaches infinity involving roots
$$\lim_{n\to\infty}\frac{n}{1+2\sqrt{n}}$$
Given my understanding of how to solve these problems, I need to take the highest power of $n$ in the denominator and then divide both the numerator and ...
18
votes
4answers
610 views
Find $\lim\limits_{n \rightarrow \infty}\dfrac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}$
Find$$\lim_{n \rightarrow \infty}\frac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}.$$
This is a recent exam question, which I couldn't figure out in the exam. My guess is it doesn't ...
1
vote
2answers
18 views
Limit with variable: non-defined expression
I have a given limit that depends on a variable $a$:
$$\lim_{x \rightarrow \infty} \left (\frac{e^{ax}}{1 - ax} \right)$$
I understand cases for $a < 0 \implies \lim = 0$ and $a > 0 \implies ...
3
votes
5answers
199 views
Why isn't this limit equal to $0$?
$f(2)=4$, $g(2)=9$, $f'(2)=g'(2)$.
$ \displaystyle \lim_{x \to 2} \frac{ \sqrt{f(x)}-2} { \sqrt{g(x)}-2} $.
Why isn't this limit equal to $0$? Since $f$ and $g$ are differentiable at $x=2$, that ...
2
votes
4answers
61 views
Finding $\lim_{x \to 0}\frac{\tan x-x}{x^3}$
Feeling like i did this wrong
$\displaystyle \lim_{x \to 0}\frac{\tan x-x}{x^3}$ $\to$ $\displaystyle \lim_{x \to 0}\frac{\sec^2x-1}{3x^2}$
$\displaystyle \lim_{x \to 0}\frac{2\tan x\sec^2x}{4x}$ ...
4
votes
3answers
98 views
How to prove that $\lim\limits_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0\text { ? }$
Let $f:\left[a,b\right]\to\mathbb{R}$ be a function that is derivative so that $f'$ is continuous then
$$
\lim_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0
$$
My attempt: I ...
1
vote
3answers
107 views
For $\lim_{x \rightarrow a}f(x) = L$, if $f(x)$ is not a constant, is $\delta(\epsilon)$ always a monotonically increasing function?
Alternatively, how does the definition of a limit guarantee that if $f(x)$ is not a constant, then a small $\epsilon$ will give me a small $\delta$?
2
votes
1answer
42 views
Problem in limits
Suppose f(x) is a differentiable function and $$\lim_{x \to \infty}{f'(x)} = 0. \\ \text{If } g(x) = f(x+1) - f(x), \text{ show that } \lim_{x \to \infty}{g(x)} = 0.$$
Does anyone have any idea?
11
votes
6answers
275 views
Why is the definition of “limit” difficult to understand at first?
Tomorrow I teach my students about limits of sequences. I have heard that the definition of limit is often difficult for students to understand, and I want to make it easier. But first I need to ...
3
votes
1answer
53 views
Limits with sums and integrals
It's one of my homework exercises that is rather problematic to me. Apparently the last thing to do is to squeeze it but I don't see yet how to do that. Could you help?
...
0
votes
1answer
28 views
Numerically evaluating functions in the limit $\to 0$
Looking for some pointers as to the rules around numerically evaluating a function in the limit e.g.
$$
f(x) = \lim_{y\to0} g(x,y) / h(y)
$$
I have seen methods where, for a series of fixed values of ...
0
votes
2answers
65 views
How can I find $\lim_{n\to \infty} a_n$
Let $$a_n=\left(1-\dfrac{1}{\sqrt2}\right)\dots \left(1-\dfrac{1}{\sqrt{n+1}}\right),n\ge1$$ Then find $\lim_{n\to \infty} a_n$.
How can I proceed? I am stuck at the first step. Please help.
0
votes
2answers
40 views
Limit of series - exponential series
Series nd continuous functions
Question :
For $0<x<\infty ,$ let $$f(x) = \sum_{0}^{\infty } e^{-nx}$$ .Show that f(x) is continuous function.
Work Done:
I know every concept of sequences and ...
1
vote
2answers
58 views
Steps to find limit as $t\to0$
So, it's been about 6 years since I've taken any type of math course, and 7 since I've taken Calculus. Recently, though, I've been trying to relearn Calculus in my free time.
I've been working ...
2
votes
5answers
102 views
Evaluate $ \lim\limits_{x \to 0} \frac{1}{\sin^3x}\int_0^x{\sin(t^2) } dt$
$$ \lim_{x \to 0} \frac{1}{\sin^3x}\int_0^x{\sin(t^2) dt}$$
This is what I've tried:
Let $F(x) = \displaystyle\int_0^x{\sin(t^2) dt}$, and let $f(x) = {\sin(t^2)}$.
Then $F'(x) = f(x) ...
11
votes
3answers
178 views
How to find the limit of these sequences?
Let $\{a_n\}$ be a real-valued sequence such that $a_1 \geq 0$ and $$a_{n+1}=\ln(a_{n}+1)$$ for all $n\ge1$. How can we find the following limits?
$$\lim_{n\to \infty}na_n=?,$$ $$\lim_{n\to ...
1
vote
3answers
156 views
How did Euler and Bernoulli prove this limit?
Prove that the lim as x approaches infinity of $(1+1/x)^x$ exists, and prove this without assuming any prior knowledge of $e$.
3
votes
3answers
106 views
Is there a way to compute $\lim\limits_{x\to\pi/3}\frac{\sqrt{3+2\cos x}-2}{\ln(1+\sin3x)}$ without using L'hopital?
I can compute
$$\lim_{x\to\pi/3}\frac{\sqrt{3+2\cos x}-2}{\ln(1+\sin3x)}$$ using L'hopital and the limit equals $\frac{\sqrt{3}}{12}$, but is there another way to compute this limit without using ...
0
votes
3answers
71 views
How to show $\lim_{n\to\infty}\frac{(2n+1)!!}{2^nn!(2n-1)}=0$?
I have a question: show
$$ \lim_{n\to\infty}\frac{(2n+1)!!}{2^nn!(2n-1)}=0. $$
Thank you in advance.
0
votes
2answers
29 views
Asymptotic behaviour of an integral
Let $I(r)$ be
$$\frac{1}{4\pi^2 r}\int_{m}^{\infty} d\rho \frac{\rho e^{-\rho r}}{\sqrt{\rho^2-m^2}}$$
How can I show that
$$I(r) \sim e^{-mr} $$ for $r\to\infty$
1
vote
3answers
45 views
Composition of Continous function is continuous
I have an example, prove that the function y = |cosx| is continuous.
We can make two function viz. let g(x) = |x| f(x) = cosx
As we know that |x| is continuous function and cosx is also continuous ...
0
votes
1answer
39 views
Limit question difference between $n^2/n$ and $n$ as $n \rightarrow \infty$
Here is a possibly VERY naive question about limits from someone who doesn't know much calculus. What is the difference between the limit of $n$ as $n \rightarrow \infty$ and the limit of ...
10
votes
8answers
232 views
Limit of $\frac{\log(n!)}{n\log(n)}$ as $n\to\infty$.
I can't seem to find a good way to solve this.
I tried using L'Hopitals, but the derivative of $\log(n!)$ is really ugly. I know that the answer is 1, but I do not know why the answer is one.
Any ...
4
votes
4answers
105 views
Limit of an integral.
$$\lim_{n\to\infty} \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^n}\,dx $$
Mathematica tells me the answer is 0, but how can I go about actually proving it mathematically?
1
vote
1answer
91 views
Can the integral of $x^x$ be found?
I'm interested in knowing if the indefinite integral of $x^x$ can be found in terms of elementary functions.
I am under the impression (be it correct or incorrect) that it can be found. This is why: ...
4
votes
4answers
133 views
Does limit means replacing $x$ for a number?
I don't understand limit so much. For example I see $\lim_{x \to -3}$. And I always just put $-3$ everywhere I see $x$. I feel like I'm doing something wrong, but it seems correct all the time.
1
vote
1answer
47 views
$\lim_{x \rightarrow \infty}\left(\frac{\pi}{2}-\tan^{-1}x\right)^{\Large\frac{1}{x}}$
Can this limit be solved without using L'Hopital's rule :
$$\lim_{x \rightarrow \infty}\left(\frac{\pi}{2}-\tan^{-1}x\right)^{\Large\frac{1}{x}}$$
Answer of this limit is : $1$
1
vote
1answer
23 views
Trigonometric limits and algebraic limit
Let $f(x) = \sin x$ , when $x \neq n\pi;$ and $2$, when $x=n\pi$ where $n \in \mathbb Z$,
$g(x) = x^2+1$, when $x\neq 2$ and equal to $~3$, when $x =2$ ,
then find $$\lim_{x\rightarrow 0}~g(f(x))$$
...
4
votes
3answers
108 views
Let $f(n)$ be the number of prime factors of the positive integer $n$. Find $\lim_{n\to \infty}\frac{f(n)} n$
Let $f(n)$ be the number of prime factors of the positive integer $n$. Find $\displaystyle \lim_{n\to \infty}\frac{f(n)} n$.
I suspect it's equal to $0$, but how can I show this? Thank you.
0
votes
1answer
73 views
Evaluating the limit: $\lim_{x\to\infty}\sqrt[x]{1+\sin(x)}$
How to evaluate the following limit?
$$\lim_{x\to\infty}\sqrt[x]{1+\sin(x)}$$
5
votes
3answers
220 views
Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?
In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one?
I can of course ...
3
votes
1answer
69 views
Derivatives using the Limit Definition
How do I find the derivative of $\sqrt{x^2+3}$? I plugged everything into the formula but now I'm having trouble simplifying.
$$\frac{\sqrt{(x+h)^2+3}-\sqrt{x^2+3}}{h}$$
0
votes
2answers
77 views
Computing $\lim_{x\to 0}[(\sin x)^{\frac{1}{x}}+(\frac{1}{x})^{\sin x}]$
How can we compute the following: $$\lim_{x\to 0}\left[(\sin x)^{\frac{1}{x}}+\left(\frac{1}{x}\right)^{\sin x}\right]?$$
The expression looks rather daunting. If $x\to 0$, then $\sin x\to 0$ but ...
1
vote
1answer
45 views
Calculating Limits Using Polar Coordinates-Phylosophical question!
I'm having difficult times understanding the rules regarding polar coordinates, in the context of calculating limits.
On the one hand, I understand that when we take $\theta$ constant, then the path ...
3
votes
2answers
71 views
limit of trigonometric function
How to find the limit of this question
$$\lim_{x \rightarrow a} \left( \frac{\sin x}{\sin a} \right)^{\frac{1}{x-a}}$$ where $a \neq k\pi$ with k an integer.
We can write this as : ...
2
votes
3answers
70 views
Show that $(x_n-y_n)$ converges to $x-y$.
Given $(x_n)$ and $(y_n)$ are sequences of real number which converge to $x$ and $y$ respectively. Show that $(x_n-y_n)$ converges to $x-y$.
If it's asking about $(x_n+y_n)$. I know that I can ...
