All Questions
95
questions
8
votes
1
answer
339
views
Prove that $\lim _{n \rightarrow \infty} \sqrt{n} \cdot\sin^{\circ n}(\frac{1}{\sqrt{n}})=\frac{\sqrt{3}}{2}$ [duplicate]
It's known that $\lim _{n \rightarrow \infty} \sqrt{n} \cdot \sin^{\circ n}(x)=\sqrt{3}$ for any $x>0$.
And I found a new conclusion $\lim _{n \rightarrow \infty} \sqrt{n} \cdot \sin^{\circ n}(\...
- 133
-2
votes
2
answers
112
views
How do I prove interval 𝐴⊂[0,3] exists on this integration
Let $f:[0,3] \to \mathbb{R}$ be a continuous function satisfying
$$\int_{0}^{3}x^{k}f(x)dx=0 \quad \text{for each k = 0,1,}\dots,n-1$$ and
$$\int_{0}^{3}x^{n}f(x)dx=3.$$
Then prove that there is an ...
- 45
4
votes
0
answers
127
views
Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
- 9,592
2
votes
0
answers
62
views
What is the precise definition of infinite limit at infinity of a function?
I saw the following definition on youtube.
Let $f$ be a function defined on some interval $(a, +\infty)$
$\lim\limits_{x \to +\infty} f(x)= +\infty$ (1)
means $\forall N > 0$ $\exists M > 0$ ...
- 33
0
votes
2
answers
99
views
What is $\lim_{x \to 0} \left(\frac{1}{e^x-1}-\frac{1}{x}\right)$ without l'Hopital's rule? [duplicate]
Any ideas on how to calculate $\lim_{x \to 0} \left(\frac{1}{e^x-1}-\frac{1}{x}\right)$ without using l'Hopital's rule?
I tried putting $u = e^x-1$ and $x = \ln(u+1)$, replacing but i dont get much ...
- 69
-1
votes
1
answer
40
views
Sequences-limit-real analysis [duplicate]
Let $f:\left[0,1\right]\to \mathbb{R}$ be a bounded function satisfying $f(2x)=3f(x)$ for $0\le x<\frac{1}{2}$
1)Show that $f(2^{n}x)=3^{n}f(x)$ for $0 \le x< \frac{1}{2^{n}}$ for all $n \in \...
- 359
0
votes
1
answer
203
views
if $f:[1, \infty)\to\mathbb{R}$, if the limit $\int^\infty _1 f(x)dx$ exists then $\lim\limits_{x\to \infty}f(x)=0$ [duplicate]
I have this:
if $f:[1, \infty)\to\mathbb{R}$ and the limit $\int^\infty _1 f(x)dx$ exists then $\lim\limits_{x\to \infty}f(x)=0$
How can I show this to be true, is it similar if it were $[0,\infty)$? ...
- 155
3
votes
1
answer
43
views
Help with the study of the function $f(x) = \frac{-2}{5x-\ln\vert x \vert}$
I'm having problems in understanding few things about this function, also because some of my calculations do not match the plot.
$$f(x) = \dfrac{-2}{5x-\ln\vert x \vert}$$
Here is what I did.
First of ...
- 3,482
0
votes
0
answers
44
views
About the definition of the limit of a function ("Mathematical Analysis 2nd Edition" by Tom M. Apostol)
4.5 LIMIT OF A FUNCTION
In this section we consider two metric spaces $(S, d_S)$ and $(T, d_T)$, where $d_S$ and $d_T$ denote the respective metrics. Let $A$ be a subset of $S$ and let $f:A\to T$ be a ...
- 8,750
2
votes
1
answer
87
views
Find for what values of a and b in R the limit exists (No De L'Hopital)
I was given this exercise in my math course at university.
The question is to find, without using De l'Hopitals and other methods which may use derivates and similars, for what values of $a$ and $b$ ...
- 91
2
votes
1
answer
47
views
Determine the parameters so that the function is continuous in R
I got this problem i've tried to solve but i don't know how to proceed.
$$\begin{cases} 3\sin(4x)&&\text{if }x\leq 0 \\ mx+q&&\text{if }x>0 \end{cases}$$
Find the value of $m$ and $...
- 93
3
votes
0
answers
56
views
Discontinuous function of two variables
Let \begin{equation}
f: \mathbb{R}^{2} \rightarrow \mathbb{R}
\end{equation} be a function of two real variables given by
\begin{equation}
f(x,y) = \begin{cases}
\frac{x}{y} & \text{...
- 61
0
votes
0
answers
61
views
Jump Continuity of the function $f(x) = \frac{ax+b}{cx+d}$
With trying with examples, I found that
the function $$f(x) = \frac{ax+b}{cx+d}$$ has a jump continuity when $a\cdot \frac{-d}{c} + b = 0$ at point
$(-d/c, \frac{a+b}{c+d})$
However, I could not find ...
- 1
0
votes
0
answers
76
views
Is my answer to this limit correct?
Define a function $ z(x, y) = \frac{11x^3+5y^3}{x^2+y^2} $. Using the continuous extension, what is the value of the following limit:
$$ \lim_{(\Delta x, \Delta y) \rightarrow (0, 0)\\ \text{along the ...
0
votes
1
answer
81
views
How can I prove that $\lim_{x\to \infty}{\sin(2x)}$ does not exits?
How can I prove that $$\lim_{x\to \infty}{\sin(2x)}$$ does not exist?
How can I prove this with the epsilon-delta definition?
- 13
0
votes
0
answers
31
views
Continuous extension of $f(x,y)=\left(\frac{1+y+x^2}{y}\right)^y$
The function $f(x,y)=\left(\frac{1+y+x^2}{y}\right)^y$ is not defined for $y=0$, so to extend $f$ continuously I want to define $f(x,0)$ as :
$$f(x,0)=\lim _{y \rightarrow 0} \left(\frac{1+y+x^2}{y}\...
- 918
1
vote
1
answer
59
views
Domain of the first derivative
I have a question about a specific exercise. Given the following function
$$f(x) = |x|sin(x^2)$$
we have that the domain is $(-\infty,+\infty)$. Now the first derivative is
$$f'(x) = \frac{xsin(x^2)}{|...
- 33
0
votes
0
answers
73
views
Find discontinuities of the function
My function is $y= \sqrt{(1-\cos(πx))/(4-x^2)}$.
The main question I have problems with answering is wherever 2 and -2 are removable discontinuities and why.
Should I think about the domain of my ...
- 103
-1
votes
1
answer
80
views
Image of $\frac{2x}{\ln(x)}$
I'm trying to calculate the image of $f(x)=\frac{2x}{\ln(x)}$. I tried to find the horizontal asymptotes. But I couldn't. the $\lim_{x \to \infty} f(x)$ is $\infty$. I also tried to find the oblique ...
- 47
6
votes
1
answer
239
views
If $a_n=\sqrt{1+\sqrt{2+\cdots\sqrt{n}}}$ and $\lim\limits_{n\to\infty}a_n=\ell$, prove $\lim_{n\to\infty}[(\ell-a_n)^{1/n}n^{1/2}]=\frac{\sqrt e}2$
$$a_n=\sqrt{1+\sqrt{2+\cdots\sqrt{n}}}$$
We can prove that $\{a_n\}$ is convergent
(using mathematical induction, $\sqrt {k+\sqrt{k+1+\cdots\sqrt{n}}}\leq k-1, for \ k\geq3$).
If
$$
\lim\limits_{n\to\...
- 783
0
votes
2
answers
76
views
Question about limits and logarithms of functions
Let $f$ be a smooth function of one-variable. Suppose that the limit $\lim_{T\rightarrow \infty}\frac{1}{T}\log f(T)$ exists. Now I am trying to see wether or not we will have that if we do a time ...
- 4,767
0
votes
0
answers
69
views
Prove that a function is constant. [duplicate]
It is an open problem (still open) published on a Chinese mathematics magazine, but i am asking this just because i can not figure it out. I wonder if someone can help? I won't use any answers here to ...
- 83
0
votes
0
answers
18
views
periodic functions, proof that lim x → + ∞ sin(x) does not exist [duplicate]
I'm trying to figure out how to proof the next thing:
Let f be a periodic function in real numbers,that is there exists T> 0 such that f (x) = f (x + T) ∀ x ∈ R. Show that if lim x → + ∞ f (x) ...
1
vote
1
answer
433
views
Show that $f(a+h)=f(a) + hf'(a+t_{h}h)$ and $\lim_{h \to 0} t_{h} = 1/2$
Let $f$ be a function of $C^{2}$ in an interval of the form $[a-h,a+h]$ where $h>0$.
Show that if $f''(a)\ne0$ then there exists $t_{h} \in [-1,1]$ such that $$f(a+h)=f(a) + hf'(a+t_{h}h) \quad, \...
- 431
0
votes
1
answer
128
views
Alternative proof of $x^x \geq \sin x$ if $x>0$
My lecturer gave me this exercise: "Show that for $x>0$ it is $x^x \geq \sin x$."
This is my approach: since $x \geq \sin x$ for all $x \geq 0$, it is sufficient to show that $x^x \geq x$;...
- 2,162
1
vote
1
answer
76
views
$\lim_{x\to\infty} (\sqrt{x+1} + \sqrt{x-1}-\sqrt{2x})$ using little o function
Problem has to be solved specifically using little o function. I was going to transform $\sqrt{x+1}$ into $1+\frac{1}{2}x+o(x)$ and $\sqrt{2x}$ into $1+\frac{1}{2}t+o(t)=1+\frac{1}{2}(2x-1)+o(2x)$ but ...
- 87
0
votes
2
answers
77
views
Find $\lim_\limits{x\to 0}\frac{1-(\cos x)^{\sin x}}{x}$ using little o
$\lim_\limits{x\to 0}\frac{1-e^{\sin(x) \ln(\cos x)}}{x}=\lim_\limits{x\to 0}\frac{1-e^{(x+o(x))\ln(1-\frac{x^2}{2}+o(x^2))}}{x}=\lim_\limits{x\to0}\frac{1-e^{(x+o(x))(-\frac{x^2}{2}+o(x^2))}}{x}$. If ...
- 87
-2
votes
1
answer
37
views
I need help with this simple problem in little-o notation [duplicate]
$\left(x-\displaystyle\frac{x^3}{6}+\displaystyle\frac{x^5}{120} +o(x^5)\right)\left(1+\displaystyle\frac{x^2}{2}-\displaystyle\frac{x^4}{24}+ \displaystyle\frac{x^4}{8}+o(x^4)\right) = x+\...
- 87
-3
votes
1
answer
53
views
Could anyone explain to me how did we get this result? This is simple example but I'm struggling with little o notation in general. [closed]
$\left(x-\displaystyle\frac{x^3}{6}+\displaystyle\frac{x^5}{120} +o(x^5)\right)\left(1+\displaystyle\frac{x^2}{2}-\displaystyle\frac{x^4}{24}+ \displaystyle\frac{x^4}{8}+o(x^4)\right) = x+\...
- 87
3
votes
2
answers
93
views
Knowing the limit of $f'(x)$ find the limit of $f(x)$
We have that $f$ is differentiable on $(a, +\infty)$ with $a>0$.
I want to show that if $\displaystyle{\lim_{x\rightarrow +\infty}f'(x)=\ell}$, then there are the following cases:
If $\ell>0$...
- 14k
0
votes
1
answer
88
views
How do I solve this limit without L'Hospital? [duplicate]
$$\lim_{x\to0}\frac{6^x-1}{x}$$
I have spent quite a lot of time on this limit but I still can't solve it. None of the regular tricks work here, I can't get rid of the $x$, nor can I get it in the ...
- 1,798
2
votes
5
answers
209
views
How do I solve this limit without l'Hopital?
I tried the substitution $t=x-(\pi/3)$ but it doesn't help at all. I have also tried using $\sin(\pi/3)=\sqrt{3}/2$ but couldn't do anything useful then. I tried to factor the denominator and ...
- 1,798
1
vote
1
answer
98
views
Continuous function and a related infimum
Let $f : [a, b) \rightarrow \mathbb{R}$ be continuous with $f(a) = 0$ and assume that there is some $x' \in ( a, b)$ such that $f(x') > 0$. Denote by
$$
x_0 := \inf \{ x \in [ a, b) : f ( x ) > ...
- 1,557
0
votes
1
answer
40
views
limit of a function defintion
Definition(limit of a function):
$f:D \subset \mathbb{R}^n \to \mathbb{R}^m$, $x_0 \in \mathbb{R}^n$ limit point of $D$
Then $ \lim_{x \to x_0}f(x)=y$ means that for each sequence $x_n \in D\...
user727154
0
votes
1
answer
66
views
Let f:(0,∞) -> R be a function such that f(1)=0, f is differentiable at 1 and f'(1)=1. Suppose that f(xy)=f(x)+f(y) for all x,y in (0,∞)
prove that lim(h->0) (f(1+h))/h = 1.
prove that f is differentiable on (0,∞) and find a formula for f'.
prove that (f(e^x)-x)'= 0 for all x in R.
I am unsure where to start with all of these ...
- 99
1
vote
2
answers
40
views
Can I determine the limit of these functions by inserting?
Given:
$$\lim_{x \rightarrow \infty}\left(\frac{ex+3}{x+e}\right)^{(x^2-1)/(2x^2+6x+4)},$$
$$\lim_{x \rightarrow 1-}(1-x)^{1-x^2}.$$
My ideas would be:
it holds that $\lim_{x \rightarrow \infty} \...
- 635
1
vote
2
answers
54
views
$ \lim_{n \to \infty} \int_0^2 f_n(x)dx$ and $ \lim_{n \to \infty} f_n(x) \text{ for } x \in [0,2]$
For $n \in \mathbb{N}$ let $f_n:[0,2] \to \mathbb{R}$ be defined by
$$f_n(x) =
\left\{
\begin{array}{ll}
n^3x^2, \text{ if } 0 \leq x \leq \frac{1}{n} \\
2n - n^2x \text{ if } \frac{1}{n} < x \...
- 931
3
votes
1
answer
75
views
Give $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $|f'(x)|<1$ $ f(x) \neq x$ for all $x \in \mathbb{R}$
Problem
Give a function $f:\mathbb{R} \rightarrow \mathbb{R}$ , $C^\infty$ such that
$1) |f'(x)|<1$
$2) f(x) \neq x$ for all $x \in \mathbb{R}$
My ideia
The idea is to get a function that ...
- 931
0
votes
1
answer
159
views
$f:[-1,1]\to \mathbb{R}$ is a continuous function such that $f(2x^2-1)=(x^3+x)f(x).$ Find $\lim_{x\to 0}\frac {f(\cos x)}{\sin x}$
$f:[-1,1]\to \mathbb{R}$ is a continuous function such that $f(2x^2-1)=(x^3+x)f(x).$ Find $\lim_{x\to 0}\frac {f(\cos x)}{\sin x}$
My approach : Using the functional equation I got f is an odd ...
- 2,269
1
vote
3
answers
60
views
Why if $\lim_{x \to a}f(x)=b$ then $\lim_{n \to \infty}f(x_n)=b$? [closed]
Why is that if $$\lim_{x \to a}f(x)=b \Rightarrow \lim_{n \to \infty}f(x_n)=b?$$
This is often used to prove that the same properties that apply to sequences, also apply to functions, but I don't ...
- 1,412
1
vote
2
answers
51
views
find limit of a multivariable function
I have to show continuity at $(0,0)$ of $f(x,y)=\frac{\sin(x^2) + \sin(y^2)}{\sqrt{x^2 +y^2}}$ for $(x,y)\ne(0,0)$ and $f(0,0)=0$.
I tried to find the limit using polar coordinates
$ \frac{\sin(r^2\...
- 41
0
votes
1
answer
69
views
Does $\lim_{x\to 0} \frac {f(x)}{|x|} \leq M $ ,for some real $M$ ,imply that $f(x)\leq |x| M $?
Just to restate my question as in the title:
Does $\lim_{x\to 0} \frac {f(x)}{|x|} \leq M $ , for some real $M$, imply that $f(x)\leq |x| M $ ?
Any help would be appreciated!
user489116
-2
votes
2
answers
45
views
$\lim\limits_{x \to 0} \frac{\sqrt{(a^4+(bx)^2} - \sqrt{a^4+b}}{x}$
calculate...
this is my homework
the photo is my solution proposal but I don't know what to do next....
- 175
0
votes
2
answers
64
views
Proof: All directional derivatives $\frac{\partial f}{\partial e}$ of $\frac{sin(x^3+y^3)}{x^2+y^2}$ are in the origin
Let $M := (0,\infty) \subset \mathbb{R^2}$ and $f:\mathbb{R}^2 \to \mathbb{R}$.
How can one prove that all directional derivatives $\frac{\partial f}{\partial e}$ of $f(x,y)$ are existing in the ...
user616397
1
vote
0
answers
26
views
Finding functions using cosine
Assume that the sine and cosine functions are continuous at the point 0.
(a) Find functions $a(x), b(h), c(x), d(h)$ such that $b(h)$ and $d(h)$ are continuous at $h = 0$ and $b(0) = d(0) = 0$, such ...
- 139
2
votes
3
answers
78
views
Find the limit of $x(x + 1 - \sin(\frac{1}{1+x})^{-1})$ as $x \rightarrow \infty$
As the title states, I need to find the limit for $x\left(x + 1 - \frac{1}{\sin(\frac{1}{1+x})}\right)$ as $x \rightarrow \infty$, as part of a larger proof I am working on.
I believe the answer is 0....
- 467
1
vote
1
answer
85
views
Does the derivative of $f(x)=\begin{cases}\dfrac{x}{2}+x^2\sin\left(\dfrac{1}{x}\right)& x\neq 0,\\0, &x=0\end{cases}$ exist everywhere?
I'm trying to prove that
$$f(x)=\begin{cases}\dfrac{x}{2}+x^2\sin\left(\dfrac{1}{x}\right)& x\neq 0,\\0, &x=0.\end{cases}$$
has a derivative everywhere. Here is what I have done:
Let $x_0\...
- 4,504
0
votes
1
answer
107
views
Product of convex functions with special properties
Let $f(x)$ and $g(x)$ be non-negative, convex functions in $C^2([M,\infty))$, where $M > 0$. Also, assume $f(x)$ is strictly decreasing on $[M,\infty)$, and that $g(x)$ is strictly increasing on $[...
- 69
6
votes
5
answers
258
views
Calculating limit $\lim\limits_{x\to\infty}\frac{3x^2-\frac{3}{x^2+1}-4f'(x)}{f(x)}$ for an unknown function.
Given that $f(x)$ is a continuous function and satisfies $f'(x)>0$ on $(-\infty,\infty)$ and $f''(x)=2 \forall x \in(0,\infty)$.We need to find the limit
$$\lim_{x\to\infty}\frac{3x^2-\frac{3}{x^...
- 241
1
vote
1
answer
151
views
Showing that Derivative is Linear
Question is: In $$f(a+h) - f(a) = h f'(a + \frac h 2), \qquad a, h \in \mathbb R$$ show $f'$ is line.
I have no problems with the first part. I'm however having trouble with taking the derivative ...
- 1,522