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Results tagged with real-analysis
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user 1021431
For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.
0
votes
1
answer
32
views
How to prove of $\lim \frac{1}{|x|+1}=0$ as $x$ goes to $-\infty$?
I understand that $\displaystyle \lim_{x\rightarrow \infty} \frac{1}{|x|+1}=0$.
Let $\varepsilon>0 $ and I choose $M>0$ such that $\frac{1}{M+1}<\varepsilon$. Thus, if $x\geq M$ then
$$\left|\frac{1}{ …
- 290
0
votes
2
answers
68
views
How to compute the norm operator of operators in $L(\mathbb R^2)$
I am trying to understand how the norm operator
$$||A||=\sup \{||Ax||: x\in \mathbb R^2\text{ with } ||x||=1\}$$
works and how to calculate it for operators in $L(\mathbb R^2)$.
If the operator is giv …
- 290
1
vote
2
answers
132
views
If $N\in \mathbb N$ and $b$ irrational, why are there finitely many rationals $r=p/q$, with ...
I am stuck with this problem:
Consider $A:=\{x\in \mathbb R : x > 0\}$. Let $b\in A$ an irrational and $N\in\mathbb N$. Then there are finitely many rationals $r=p/q$, with $p,q$ coprime and $0<q\leq …
- 290
1
vote
2
answers
30
views
Continuity of partial derivatives of $f(x,y) =\dfrac{xy(x^2-y^2)}{x^2+y^2}$ if $(x,y)\neq 0$...
So I got that the partial derivatives of
\begin{equation*}
f(x,y) =
\begin{cases}
\dfrac{xy(x^2-y^2)}{x^2+y^2} & \text{if } (x,y) \not= (0,0),\\
0 & \text{if } (x,y) = (0,0). …
- 290
1
vote
0
answers
36
views
Proving that $d(x,y)=0$ if and only if $x=y$ for a given set of continuous functions
I found the following problem:
Let $X$ be the set of continuous functions from $[a,b]$ into $\mathbb R$. For $x,y\in X$ define $d(x,y)$ by
$$d(x,y) = \int_a^b |x(t) - y(t)|\, dt.$$
Prove that $(X, d) …
- 290
5
votes
0
answers
115
views
Proving that limits at infinity are unique help
I am trying to prove that the $\displaystyle \lim_{x\rightarrow \infty}f(x)=L$ is unique, if it exists.
Definition: Let $f:S\rightarrow\mathbb R$ be a function, where $\infty$ is a cluster point of $ …
- 290
1
vote
2
answers
217
views
Prove that $1/x^2$ is not uniformly continuous on $(0,\infty)$ using $\varepsilon$-$\delta$ ...
I am trying to prove that $f(x) = 1/x^2$ is not uniformly continuous on $(0,\infty)$ using $\varepsilon$-$\delta$ arguments.
This is what I got so far.
Proof: Let $\varepsilon = 1$. For every $\delta> …
- 290
3
votes
0
answers
286
views
How to prove that $h(x):=\int_a^bg(t-x)f(t)dt$ is Lipschitz continuous?
Assume that $f:[a,b]\rightarrow \mathbb R$ is continuous and $g:\mathbb R\rightarrow \mathbb R$ is Lipschitz continuous. Now define
$$
h(x):=\int_a^b g(t-x)f(t)dt.
$$
I want to show that $h$ is Lipsch …
- 290
1
vote
2
answers
195
views
Can we compute this integral $\int_0^{2\pi} \frac{1}{5+3 \cos x} dx$ using an antiderivative... [duplicate]
I was trying to compute this integral
$$\int_0^{2\pi} \frac{1}{5+3 \cos x} dx$$
using the substitution method $t = \tan \frac{x}{2}$ suggested in Michael Spivak's book: Calculus 3rd ed., pages 382-383 …
- 290
0
votes
3
answers
126
views
Is a bounded polynomial constant? [duplicate]
I am trying this problem:
If $p(x)$ is a bounded polynomial for all $x\in \mathbb R$, then $p(x)$ must be a constant.
I am trying to prove it by contradition. So I assume that $p(x)$ is bounded for …
- 290