Tagged Questions
0
votes
1answer
59 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 ...
2
votes
6answers
123 views
Solve the following limit?
Please only give hints? $$\lim_{x \to \frac{\pi}{6}}\frac{2\sin{(x)}-1}{\sqrt{3}\tan{(x)}-1}$$ I tried this and I was able to simplify it down to $$\lim_{x \to ...
2
votes
3answers
57 views
How can I find the limit of $\lim _{x\rightarrow \infty }\left( 4x^{2}\sin ^{2}\left( \dfrac {2} {x}\right) \right)$ to infinity?
The title basically tells everything. The result is 16 but I can't figure out how to do this. Thanks!
2
votes
3answers
105 views
Limit without L'Hopital's rule
Please solve this withouth L'Hopital's rule? $$\lim_{x\rightarrow\sqrt{3}} \frac{\tan^{-1} x - \frac{\pi}{3}}{x-\sqrt{3}}$$
All I figured out how to do is to rewrite this as $$\frac{\tan^{-1} x - ...
1
vote
4answers
83 views
$\lim_{x\to 0} x^3/\tan^3(2x)$
$$\lim_{x\to 0}\frac{x^3}{\tan^3(2x)} $$
My textbook has an answering of $\frac{1}{8}$ and I'm quite confused on how they got that. Only thing that I could see to get an $8$ would be $2^3$ from ...
2
votes
2answers
64 views
$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$
$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$.
1
vote
4answers
91 views
$\lim_{x\rightarrow\infty}\sin(x)$?
In physics I came across these kind of equations when I am trying to find the asymptotic behaviour of some function.
Can anyone explain if there is any sense in talking about $\sin(x)$ or $\cos(x)$ ...
6
votes
4answers
124 views
Finding the limit of $\frac{1-\cos(2x)}{1-\cos(3x)}$ for $x \to 0$
As $x$ goes to $0$, what is the limit of
$$\frac{1-\cos(2x)}{1-\cos(3x)}$$
Thanks.
6
votes
3answers
159 views
Trigonometrical limit $\lim\limits_{ x\to 0 } \frac{\sin x - x\cos x}{x^3}?$
Can you help me solve this without using de l'Hôpital's rule (just using Standard rules):
$$ \lim_{ x\to 0 } \frac{\sin x - x\cos x}{x^3}? $$
1
vote
1answer
58 views
Why is $\lim\limits_{h\to 0}h\cos\frac1h\stackrel{?}=0 $?
Can someone explain why is this true? $$\lim_{h\to 0}h\cos\frac1h\stackrel{?}=0$$
$\lim_{h\rightarrow0}\cos{\frac{1}{h}}$ is undefined (limit does not exist), right? So how can the above be true?
12
votes
4answers
250 views
Can $\sin n$ get arbitrarily close to $1$ for $n\in\mathbb{N}?$
Or put differently, does
$$\lim_{n \to \infty}\big(\max \{\sin 1, \sin 2, \ldots ,\sin n\}\big) = 1?$$
My intuition says yes, but how can one prove this?
2
votes
2answers
96 views
$\int_{-\infty}^\infty \frac{\lambda l}{2\pi\epsilon_0(x^2+l^2)^{3/2}}dx$ Proving an electric field from a wire falls off at $1/l$
If you don't know the physics behind all this, that's okay, I just need the integral of this function (or limit, I'm not too sure).
Here's the gist: normally with infinitesimal point charges, there ...
2
votes
2answers
68 views
How can this trigonometric inequality related to a limit be proved?
I want to prove that $\;\;\displaystyle \left|\frac{\sin x-x}{x^{2}}\right|\leq\frac{4(\pi/2-1)}{\pi^{2}}\;\;$
for all $x$ such that $x\in\left[0,\pi/2\right]$.
If you look at the graph of the ...
0
votes
2answers
46 views
One sided limits for $\displaystyle\lim_{x\to\frac{\pi}{4}} \frac{1}{\cot{x}-1}$
For the limit: $\displaystyle\lim_{x\to\frac{\pi}{4}} \frac{1}{\cot{x}-1}$ the one sided limits are $\displaystyle\lim_{x\to\frac{\pi}{4}^{-}} \frac{1}{\cot{x}-1}=\infty$ and ...
3
votes
5answers
140 views
Derivation of the limit $\sin x/x$
Reference: How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
I think that all proofs above are beautiful but I cannot see why we have to complicate it. We know that for small values of $x$ that ...
7
votes
4answers
212 views
Strictly formal proof of $ \displaystyle \lim_{x \to 0} \frac{\sin(x)}{x} = 1 $.
I’m looking for a proof of $ \displaystyle \lim_{x \to 0} \frac{\sin(x)}{x} = 1 $ that does not use other trigonometric functions or any first-order approximation to the sine function. Is this ...
4
votes
1answer
169 views
Does $\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(…\cos x))}}$ exist? [duplicate]
Possible Duplicate:
Explaining $\cos^\infty$
Does the following limit exist?
$$\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(...\cos x))}}$$
If yes, find the limit.
If no, please ...
1
vote
4answers
62 views
Manipulating trig limit functions
I'm having difficulty understanding how my calc teacher manipulated this problem
What I dont understand is how he minipulated the second line to the third line. In other words, I don't understand ...
4
votes
3answers
129 views
What is the solution for $\lim\limits_{m\to\infty}\left(\cos\frac xm\right)^{m}$?
Please, help me in solving of $\lim\limits_{m\to\infty}\left(\cos\frac xm\right)^{m}$.
3
votes
4answers
145 views
What is limit of: $\displaystyle\lim_{x\to 0}$$(\tan x - \sin x)\over x$
I want to search limit of this trigonometric function:
$$\displaystyle\lim_{x\to 0}\frac{\tan x - \sin x}{x^n}$$
Note: $n \geq 1$
5
votes
3answers
112 views
prove without using L'Hopital
I was asked to prove this , without using L'Hopital... tried out some trig. identities with no big use $(\sin(\alpha)-\sin(\beta))(\sin(\alpha)+\sin(\beta))=\sin^2(\alpha)-\sin^2(\beta)$ for example, ...
1
vote
2answers
111 views
Limit of a trigonometric function
I'm trying to find the following limit:
$$
\lim_ {x \to 1} \frac{\sin{\pi x}}{1 - x^2}
$$
I can't figure it out how to reach the fundamental trigonometric limit. Everything i see is that the ...
1
vote
2answers
127 views
Limit of trigonometric functions using identities.
I would like to solve the following using only trig identities.
$$
\lim_{x \to \pi} {\cot2(x-\pi)}{\cot(x-\frac\pi2)}
$$
I have so far that the above is equal to
$$
\lim_{x \to \pi} ...
4
votes
2answers
151 views
Finding a limit without series expansion and l'Hopital's rule
I have to find
$$\lim_{x\to 0}\frac{\tan x-\sin x}{\sin^3x}$$
without series expansion nor l'Hopital's rule, and am utterly and completely lost.
I ended up putting $x = 2y$ and getting to
...
4
votes
4answers
148 views
How to prove $\lim_{n \to +\infty} \sqrt{n}\int_0^\pi{\cos(\frac{t}{2})^n}dt>0$
I want to prove
$$\lim_{n \to +\infty}\sqrt{n}\int_0^\pi{\cos\left(\frac{t}{2}\right)^n}dt>0.$$
First, I consider $$\lim_{n \to ...
5
votes
4answers
208 views
Explaining $\cos^\infty$
I noticed something odd while messing around on my calculator.
$$\lim_{x\to \infty} \cos^x(c)=0.7390851332$$ Where $c$ is a real constant.
My calculator is in radians and I got this number by simply ...
2
votes
2answers
128 views
What is the limit of this function as $n$ tends to infinity?
$\lim_{n\rightarrow\infty}\sqrt{n}\cos(\frac{\pi}{2}-\frac{1}{\sqrt{n}})$
I'm having a lot of trouble figuring it out.
My first step is always to convert $\cos(\frac{\pi}{2}-\frac{1}{\sqrt{n}})$ to ...
1
vote
2answers
408 views
Find the limit without l'Hôpital's rule
Find the limit
$$\lim_{x\to 1}\frac{(x^2-1)\sin(3x-3)}{\cos(x^3-1)\tan^2(x^2-x)}.$$
I'm a little rusty with limits, can somebody please give me some pointers on how to solve this one? Also, ...
3
votes
2answers
333 views
limit of inverse trigonometric function
I have the following limit that I am trying to solve but apparently I am stuck in doing l'Hôpital's rule and going nowhere so any help would be appreciated
$$\lim_{x\to 0} \frac{\arcsin ...
3
votes
5answers
620 views
Sequence of solutions to $x\sin x=1$
Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed.
Consider a sequence $x_n, n\ge1$ formed by positive solutions to ...
3
votes
1answer
72 views
How can I solve this limit?
$\lim_{x\to 1} \frac{\sin (x-1)}{x-1}$
I know the answer equals $1$ because $\lim_{x\to 0} \frac{\sin (x)}{x} = 1$ and in the following question $x-1$ gets arbitrary close to 0 so the same thing is ...
2
votes
2answers
217 views
calculating limit without using direct formulae
In case of $\delta$-$\epsilon$ definition of limit, books usually show that some $L$ is the limit of some function and then they prove it by reducing $L$ from the function $f(x)- L$ and showing ...
1
vote
1answer
64 views
How to find the limit of $\sin(f(x))$, given the graph of $f(x)$
The full question is uploaded here: http://imgur.com/EZekb
Basically, given the graph shown in the image above, I thought that the limit of $\sin(f(x))$ would be $\sin(2)$, since the limit of just ...
9
votes
3answers
151 views
Rigorous way to find the limit of this difference?
This is a question from an old released exam.
By the triangle inequality, $s-r<1$, so I eliminate answers D and E. Intuitively, since the lower angle between $1$ and $r$ is fixed at $110^\circ$, ...
6
votes
7answers
312 views
Is it necessary to have $\theta$ in radians to obtain $\frac{\sin{\theta}}{\theta} \to 1$ as $\theta\to 0$?
$$\lim_{\theta \to 0} \frac{\sin{\theta}}{\theta} = 1$$
Could anyone explain why $\theta$ is in radians?
Is it because if $\theta$ is not in radians then $\frac{\sin{\theta}}{\theta} \neq 1$?
Is ...
1
vote
2answers
158 views
What is the answer for the $\lim\limits_{n\rightarrow \infty} \frac{\sin(nt)}{\sin(t)}$?
Let $t\in (0,\pi)$ and $n$ change in natural numbers. I am wondering what is the answer to the following limit.
$$\lim_{n\rightarrow \infty} \frac{\sin(nt)}{\sin(t)}.$$
Thank you.
4
votes
1answer
292 views
Finding Limits of Trig Functions
I am asked find the following limit
$$\lim_{\theta \rightarrow 0}\frac {\sin^2\theta}{\theta}$$
I recognize that $$\lim_{\theta \rightarrow 0}\frac{\sin\theta}{\theta}=1$$
But because I have ...
1
vote
4answers
155 views
How can I calculate $\lim_{x \to 0} \log(\cos(x))/\log(\cos(3x))$ without l'Hopital?
How can I calculate the following limit without using, as Wolfram Alpha does, without using l'Hôpital?
$$
\lim_{x\to 0}\frac{\log\cos x}{\log\cos 3x}
$$
4
votes
1answer
65 views
Evaluating $\lim_{y \to 0^+} (\cosh (3/y))^y$
Evaluating $$\lim_{y \to 0^+} (\cosh (3/y))^y$$
This is what I have tried:
$L = (cosh(3/y))^y$
$\ln L = \frac{\cosh(3/y)}{1/y}$, applying L'Hopital's rule, I get:
$\ln L = \frac{-3y^2(\sinh ...
7
votes
3answers
218 views
how to find this limit $\lim_{x\rightarrow 0} \frac{\sin x^2}{ \ln ( \cos x^2 \cos x + \sin x^2 \sin x)} = -2$ without using L'Hôpital's rule
I am looking for simple trigonometric or algebraic manipulation so that this limit can be solved without using L'Hôpital's rule
$$ \lim_{x\rightarrow 0} \frac{\sin x^2}{ \ln ( \cos x^2 \cos x + \sin ...
1
vote
4answers
152 views
why $\lim_{x\to-\infty}(\sin x+2)\ln(-x)=\infty$?
Why does $\lim_{x\to-\infty}(\sin x+2)\ln(-x)$ equal $\infty$?
Breaking up the limit:
$\lim_{x\to-\infty}(\sin x+2)$ DNE because it oscillates between 1 and 3
$\lim_{x\to-\infty}\ln(-x) = \infty$
...
10
votes
3answers
211 views
$\cos(x)$ and $\arccos(x)$ couple limit
Find the value of the following limit:
$$\lim_{n\to\infty} \frac {\cos 1 \cdot \arccos \frac{1}{n}+\cos\frac
{1}{2} \cdot \arccos \frac{1}{(n-1)}+ \cdots +\cos \frac{1}{n} \cdot
...
7
votes
1answer
373 views
How to turn this sum into an integral?
I have been trying to find the closed form of this sum to no avail. It was suggested to me to try and turn this sum into an integral and solve it like that. However, I am confused as to how to do ...
2
votes
1answer
225 views
Convergence and closed form of this infinite series?
If we have a circle of radius $r$ with an $n$-gon inscribed within this circle (i.e. with the same circumradius), we can find the difference of the areas using:
$$A_n =\overbrace{\pi r^2}^\text{Area ...
1
vote
1answer
256 views
Showing that $\cos n\pi\theta$ is periodic unless $\theta$ is an even integer.
I am trying to show that $\cos n\pi\theta$ is periodic unless $\theta$ is an even integer.
I wish to provide a proof based on an example of the $\sin n\pi\theta$ case of the first result, I would ...
2
votes
5answers
197 views
Why $x<\tan{x}$ while $0<x<\frac{\pi}{2}$?
In proof of $\displaystyle\lim_{x\rightarrow0}\frac{\sin{x}}{x}=1$ is assumed that $\sin{x}\leq{x}\leq\tan{x}$ while $0<x<\frac{\pi}{2}$. First comparison is clear, arc length must be greater ...
3
votes
2answers
296 views
Wrong initial intuition about the limit of $\frac{\sin(x)}{x}$ as $x$ goes to 0
I'm following the MIT OpenCourseware course on calculus, and when proving the derivative of $\sin(x)$ the following assumption is needed
$\lim\limits_{x\to0} \frac{\sin(x)}{x} = 1$
The proof for ...
4
votes
4answers
305 views
Why is $\lim\limits_{x \space \to \infty}\space{\arctan(x)} = \frac{\pi}{2}$?
As part of this problem, after substitution I need to calculate the new limits.
However, I do not understand why this is so:
$$\lim_{x \to \infty}\space{\arctan(x)} = \frac{\pi}{2}$$
I tried ...
4
votes
2answers
139 views
Trigonometric limit
In order to prove that $\displaystyle\lim_{x \to 0}\frac{1-\cos(ax)}{ax}=0$, with $a \ne 0$, I managed that $a=2$ and evaluated this limit:
$$ \begin{align*} \quad \lim_{x \to ...
0
votes
3answers
309 views
A limit involving inverse cosine function
Let $$f(a,b,\lambda,\gamma)=\frac{1}{2}[(1+\cos(\gamma))\cos(\lambda(b-a))+(1-\cos(\gamma))\cos(\lambda(b+a))],$$ for $a\ge 0, b\ge 0$.
I am sure that
$$\lim_{(a,b)\to (0,0)} ...
