Question

Evaluate $\lim_{x\rightarrow \infty}\frac{\ln \left( \sqrt[x]{a_1^x+a_2^x+...+a_n^x}-a_1 \right)}{x}$

score 163 · Answer 1 · 2026-04-17 11:59:04

163

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Evaluate $\lim_{x\rightarrow \infty}\frac{\ln \left( \sqrt[x]{a_1^x+a_2^x+...+a_n^x}-a_1 \right)}{x}$

Published on 17 Apr 2026 - 11:59

score 1.1k · Answer 2 · 2026-04-29 04:48:34

1.1k

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How to get the upper and lower bounds of a graph?

Published on 29 Apr 2026 - 4:48

score 51 · Answer 3 · 2026-04-19 21:43:47

51

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Given $f(x) = 3x^4-2x^3-x^2 +1, \ g(x) = 4x^3-3x^2-2x$ show $\frac{f(1)-f(0)}{g(1) - g(0)} \ne \frac {f'(x)}{g'(x)}$ for any $x \in (0,1)$

Published on 19 Apr 2026 - 21:43

score 63 · Answer 4 · 2026-04-16 03:56:09

Counterexamples to "If $\lim_{x\to \infty} f(x)=0$, then for large enough $x$,$f(x)$ gets closer and closer to $0$, without ever equalling $0$."

score 193 · Answer 5 · 2026-04-15 17:07:01

193

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Baffled with calculating $\lim\limits_{x\to +\infty}{(2xe^x-2e^x+1)}$

Published on 15 Apr 2026 - 17:07

score 4.2k · Answer 6 · 2026-04-22 11:42:47

4.2k

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Prove that $\lim_{x\rightarrow 0}\frac{f(x^2)-f(0)}{x}=0$ if $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable at $x=0$

Published on 22 Apr 2026 - 11:42

score 52 · Answer 7 · 2026-04-11 21:32:15

52

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Proving existence of following integral

Published on 11 Apr 2026 - 21:32

score 95 · Answer 8 · 2026-04-15 10:15:19

95

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Computing the limit $\lim_{x\to \infty } \frac{x^2(1+\sin^2(x))}{(x+\sin(x))^2} $

Published on 15 Apr 2026 - 10:15

score 205 · Answer 9 · 2026-04-15 03:59:06

205

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Baffled with $\lim\limits_{x\to 0}{e^x-e^{\sin x} \over x-\sin x}$

Published on 15 Apr 2026 - 3:59

score 55 · Answer 10 · 2026-04-16 06:48:21

55

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How to negate $\forall\epsilon > 0,\; \exists n_0=n_0(\epsilon)\in \Bbb{N},\ni |a_n-a^{*}| <\epsilon,\;\;n\geq n_0$

Published on 16 Apr 2026 - 6:48

score 28 · Answer 11 · 2026-04-20 06:00:14

28

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Proving a particular identity for differentiable functions.

Published on 20 Apr 2026 - 6:00

score 71 · Answer 12 · 2026-04-28 20:25:27

71

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Troubles with a limit sequence.

Published on 28 Apr 2026 - 20:25

score 64 · Answer 13 · 2026-04-16 11:25:05

64

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Why this :$ \lim_{n\to \infty } n (\frac{1+i}{2})^n=0$?

Published on 16 Apr 2026 - 11:25

score 728 · Answer 14 · 2026-04-12 12:13:19

728

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Definition of a derivative

Published on 12 Apr 2026 - 12:13

score 72 · Answer 15 · 2026-04-17 15:47:59

72

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Limit of the logarithm of values very close to one: $\frac{\ln(1-\epsilon_A^2)}{\ln(1-\epsilon_B^2)}$

Published on 17 Apr 2026 - 15:47

List Question

Evaluate $\lim_{x\rightarrow \infty}\frac{\ln \left( \sqrt[x]{a_1^x+a_2^x+...+a_n^x}-a_1 \right)}{x}$

How to get the upper and lower bounds of a graph?

Given $f(x) = 3x^4-2x^3-x^2 +1, \ g(x) = 4x^3-3x^2-2x$ show $\frac{f(1)-f(0)}{g(1) - g(0)} \ne \frac {f'(x)}{g'(x)}$ for any $x \in (0,1)$

Counterexamples to "If $\lim_{x\to \infty} f(x)=0$, then for large enough $x$,$f(x)$ gets closer and closer to $0$, without ever equalling $0$."

Baffled with calculating $\lim\limits_{x\to +\infty}{(2xe^x-2e^x+1)}$

Prove that $\lim_{x\rightarrow 0}\frac{f(x^2)-f(0)}{x}=0$ if $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable at $x=0$

Proving existence of following integral

Computing the limit $\lim_{x\to \infty } \frac{x^2(1+\sin^2(x))}{(x+\sin(x))^2} $

Baffled with $\lim\limits_{x\to 0}{e^x-e^{\sin x} \over x-\sin x}$

How to negate $\forall\epsilon > 0,\; \exists n_0=n_0(\epsilon)\in \Bbb{N},\ni |a_n-a^{*}| <\epsilon,\;\;n\geq n_0$

Proving a particular identity for differentiable functions.

Troubles with a limit sequence.

Why this :$ \lim_{n\to \infty } n (\frac{1+i}{2})^n=0$?

Definition of a derivative

Limit of the logarithm of values very close to one: $\frac{\ln(1-\epsilon_A^2)}{\ln(1-\epsilon_B^2)}$

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