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\...
- 9,534
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
98
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 \...
- 319
0
votes
1
answer
201
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 ...
- 8,740
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