Question

If $f \in C([0, \infty))$ and $\lim_{x\rightarrow\infty} f(x)\int_0^x f^2(t) \ dt = 1$, then $f(x) \sim \left(\frac{1}{3x}\right)^{1/3}$

score 152 · Answer 1 · 2021-03-30 08:43:25

152

Views

If $f \in C([0, \infty))$ and $\lim_{x\rightarrow\infty} f(x)\int_0^x f^2(t) \ dt = 1$, then $f(x) \sim \left(\frac{1}{3x}\right)^{1/3}$

Published on 30 Mar 2021 - 8:43

score 51 · Answer 2 · 2021-03-31 14:31:02

51

Views

How to prove that $u(\epsilon') = \inf\{f(s) - f(s+\epsilon'): s \in [0,t]\} > 0$ for strictly decreasing function $f$?

Published on 31 Mar 2021 - 14:31

score 70 · Answer 3 · 2021-03-31 21:46:52

70

Views

$\delta-\varepsilon$ definition of $\lim_{x \to \infty} f(x) = \infty$

Published on 31 Mar 2021 - 21:46

score 76 · Answer 4 · 2026-04-04 21:04:28

76

Views

Uniform continuity of function v continuity

Published on 04 Apr 2026 - 21:04

score 67 · Answer 5 · 2021-04-02 05:38:58

67

Views

$\frac{1}{A}e^{(c-\varepsilon)\sqrt{n}} < g(n) < e^{c\sqrt{n}}$ implies $\log{g(n)} \sim c\sqrt{n}$.

Published on 02 Apr 2021 - 5:38

score 54 · Answer 6 · 2021-04-02 11:16:13

54

Views

How do I use epsilon delta definition for this problem?

Published on 02 Apr 2021 - 11:16

score 124 · Answer 7 · 2021-04-03 19:08:53

124

Views

Validity of an epsilon delta proof.

Published on 03 Apr 2021 - 19:08

score 75 · Answer 8 · 2021-04-04 11:19:34

75

Views

On the definition of continuity.

Published on 04 Apr 2021 - 11:19

score 123 · Answer 9 · 2021-04-05 19:32:12

123

Views

Proof of quotient rule for limits, is it correct?

Published on 05 Apr 2021 - 19:32

score 60 · Answer 10 · 2021-04-06 15:12:38

60

Views

Show that $\lim_{x\to a}(f(x)-g(x))=+\infty$

Published on 06 Apr 2021 - 15:12

score 124 · Answer 11 · 2021-04-06 16:02:57

124

Views

Show that $h$ is a constant.

Published on 06 Apr 2021 - 16:02

score 52 · Answer 12 · 2021-04-06 21:25:38

52

Views

A question about epsilon delta continuity.

Published on 06 Apr 2021 - 21:25

score 171 · Answer 13 · 2021-04-07 09:57:50

171

Views

Formal definition of limit (epsilon delta)

Published on 07 Apr 2021 - 9:57

score 63 · Answer 14 · 2021-04-07 10:30:33

63

Views

Please check if there is anything wrong with this proof.

Published on 07 Apr 2021 - 10:30

score 153 · Answer 15 · 2021-04-07 19:47:06

153

Views

Epsilon-Delta proof of multivariable function

Published on 07 Apr 2021 - 19:47

List Question

If $f \in C([0, \infty))$ and $\lim_{x\rightarrow\infty} f(x)\int_0^x f^2(t) \ dt = 1$, then $f(x) \sim \left(\frac{1}{3x}\right)^{1/3}$

How to prove that $u(\epsilon') = \inf\{f(s) - f(s+\epsilon'): s \in [0,t]\} > 0$ for strictly decreasing function $f$?

$\delta-\varepsilon$ definition of $\lim_{x \to \infty} f(x) = \infty$

Uniform continuity of function v continuity

$\frac{1}{A}e^{(c-\varepsilon)\sqrt{n}} < g(n) < e^{c\sqrt{n}}$ implies $\log{g(n)} \sim c\sqrt{n}$.

How do I use epsilon delta definition for this problem?

Validity of an epsilon delta proof.

On the definition of continuity.

Proof of quotient rule for limits, is it correct?

Show that $\lim_{x\to a}(f(x)-g(x))=+\infty$

Show that $h$ is a constant.

A question about epsilon delta continuity.

Formal definition of limit (epsilon delta)

Please check if there is anything wrong with this proof.

Epsilon-Delta proof of multivariable function

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