Question

Let $L=\lim_{x\to 0} \frac{ a-\sqrt {a^2-x^2} -\frac{x^2}{2}}{x^4}$, $a>0$. If $L$ is finite, find $a$ and $L$

score 195 · Answer 1 · 2020-08-16 15:39:25

195

Views

Let $L=\lim_{x\to 0} \frac{ a-\sqrt {a^2-x^2} -\frac{x^2}{2}}{x^4}$, $a>0$. If $L$ is finite, find $a$ and $L$

Published on 16 Aug 2020 - 15:39

score 150 · Answer 2 · 2020-08-17 18:49:56

150

Views

Finding $\lim_{x \to \infty} (x + \frac{2x^{3}}{3} - \frac{2(x^2+1)^{\frac{3}{2}}}{3})$

Published on 17 Aug 2020 - 18:49

score 129 · Answer 3 · 2020-08-20 19:32:25

Limit without de l'Hôpital: $\lim _{x\to 0\color{red}{\boldsymbol -}}\left(1+x^3\right)^{\frac{1}{\left(x^2+1\right)^4-1}}$

score 151 · Answer 4 · 2020-08-21 12:32:20

151

Views

Solve $\lim_{x\rightarrow +\infty}\frac{1}{x}\log\left(\frac{x+1}{1+x^2}\right) = 0$ without L'Hôpital's rule

Published on 21 Aug 2020 - 12:32

score 66 · Answer 5 · 2020-08-26 19:12:28

66

Views

Limit $\lim_{x\to 0}(1-x)^{\frac{\log(1+\frac{x^2}2)}{\sin^5x}}$ without the use of the Hôpital's rule and asymptotics $\sim_0$

Published on 26 Aug 2020 - 19:12

score 312 · Answer 6 · 2020-08-27 11:11:40

312

Views

Ambiguity in limits

Published on 27 Aug 2020 - 11:11

score 101 · Answer 7 · 2020-08-28 00:30:19

101

Views

How to simplify $ \lim_{x \to 0+}\frac{4e^{x^2}-4\cos(x^3)-2\sin(x^4)-8\sqrt{1+x^2}-x^4+8}{1-\sqrt[3]{1+13x^6}}$ to find the solution

Published on 28 Aug 2020 - 0:30

score 836 · Answer 8 · 2020-08-28 11:24:33

836

Views

How to evaluate $\lim _{x\to 2^+}\left(\frac{\sqrt{x^2-4}+\sqrt{x}-\sqrt{2}}{\sqrt{x-2}}\right)$ (without L'Hopital)?

Published on 28 Aug 2020 - 11:24

score 100 · Answer 9 · 2026-04-04 03:21:39

100

Views

Finding the limit: $\lim_{x\to \infty}\frac{1}{2}x\sin {\frac{180(x-2)}{x}}$

Published on 04 Apr 2026 - 3:21

score 138 · Answer 10 · 2020-08-29 19:56:06

138

Views

$δ,ε$-proof for limit $\frac{\sqrt{2-2\cos(x)}}{x}$.

Published on 29 Aug 2020 - 19:56

score 63 · Answer 11 · 2020-08-30 07:32:09

63

Views

How to prove that $f: \mathbb{R} \to \mathbb{R}$ is continuous, given that $xy-y^2 \leq f(x)-f(y) \leq x^2-xy$?

Published on 30 Aug 2020 - 7:32

score 182 · Answer 12 · 2020-08-31 10:02:29

182

Views

Splitting a limit in two variables

Published on 31 Aug 2020 - 10:02

score 87 · Answer 13 · 2020-08-31 15:49:18

87

Views

Proving for infinite limits when $x$ tends to $a$.

Published on 31 Aug 2020 - 15:49

score 102 · Answer 14 · 2020-09-02 19:29:41

102

Views

Verification for a series of limits

Published on 02 Sep 2020 - 19:29

score 212 · Answer 15 · 2020-09-03 05:04:35

212

Views

Find the limit $\lim_{x\to0}\left(\frac{1}{\arcsin x}-\frac{1}{\sin x}\right)$

Published on 03 Sep 2020 - 5:04

List Question

Let $L=\lim_{x\to 0} \frac{ a-\sqrt {a^2-x^2} -\frac{x^2}{2}}{x^4}$, $a>0$. If $L$ is finite, find $a$ and $L$

Finding $\lim_{x \to \infty} (x + \frac{2x^{3}}{3} - \frac{2(x^2+1)^{\frac{3}{2}}}{3})$

Limit without de l'Hôpital: $\lim _{x\to 0\color{red}{\boldsymbol -}}\left(1+x^3\right)^{\frac{1}{\left(x^2+1\right)^4-1}}$

Solve $\lim_{x\rightarrow +\infty}\frac{1}{x}\log\left(\frac{x+1}{1+x^2}\right) = 0$ without L'Hôpital's rule

Limit $\lim_{x\to 0}(1-x)^{\frac{\log(1+\frac{x^2}2)}{\sin^5x}}$ without the use of the Hôpital's rule and asymptotics $\sim_0$

Ambiguity in limits

How to simplify $ \lim_{x \to 0+}\frac{4e^{x^2}-4\cos(x^3)-2\sin(x^4)-8\sqrt{1+x^2}-x^4+8}{1-\sqrt[3]{1+13x^6}}$ to find the solution

How to evaluate $\lim _{x\to 2^+}\left(\frac{\sqrt{x^2-4}+\sqrt{x}-\sqrt{2}}{\sqrt{x-2}}\right)$ (without L'Hopital)?

Finding the limit: $\lim_{x\to \infty}\frac{1}{2}x\sin {\frac{180(x-2)}{x}}$

$δ,ε$-proof for limit $\frac{\sqrt{2-2\cos(x)}}{x}$.

How to prove that $f: \mathbb{R} \to \mathbb{R}$ is continuous, given that $xy-y^2 \leq f(x)-f(y) \leq x^2-xy$?

Splitting a limit in two variables

Proving for infinite limits when $x$ tends to $a$.

Verification for a series of limits

Find the limit $\lim_{x\to0}\left(\frac{1}{\arcsin x}-\frac{1}{\sin x}\right)$

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