8 votes
Accepted

If $\lim\limits_{x\to0}f(x)=0$ and $\lim\limits_{x \to 0}\frac{f(2x)-f(x)}{x}=0$ how to rigorously prove that $\lim\limits_{x \to 0}\frac{f(x)}{x}=0$?

Since $\lim_{x \rightarrow 0} f(x) = 0$ you can write $f(x)$ as a telescoping sum $$f(x) = \sum_{n=0}^{\infty} (f(2^{-n} x) - f(2^{-n-1}x))$$ $$ = \sum_{n=0}^{\infty} (2^{-n - 1}x) {f(2^{-n} x) - f(2^{...
Zarrax's user avatar
  • 44.8k
5 votes

How to solve this limit without using L'Hopital

If you are comfortable with differentiation limits this is just $\tan''(a)=2\sec^2(a)\tan(a)$ But a solution can also be done with pure trig this would ofc mirror the process of finding the derivative ...
RandomGuy's user avatar
  • 1,044
5 votes
Accepted

How to solve this limit without using L'Hopital

By a simple but long way, we can use that $$\tan(a+2h)=\frac{\tan a+\tan 2h}{1-\tan a\tan 2h}, \;\;\;\tan(a+h)=\frac{\tan a+\tan h}{1-\tan a\tan h}$$ and then $$\frac{\tan(a+2h)-2\tan(a+h)+\tan a}{h^2}...
user's user avatar
  • 154k
4 votes

With $x_1=1$ and $x_{n+1} = \frac{1}{x_1^2+x_2^2+\dots+x_{n}^2}$show$\lim\limits_{n\to\infty}\frac{x_1+x_2+\dots+x_n}{n^{2/3}}=\frac{\sqrt[3]{9}}{2}$

This is my first time answering question on this website.If my editing form is wrong,please forgive me.Thanks. $$ x_{n+1}=\frac{1}{{x_1}^2+{x_2}^2+...+{x_{n-1}}^2+{x_n}^2} \\ so\,\,x_n=\frac{1}{{x_1}^...
megumin's user avatar
  • 41
4 votes
Accepted

Confusion about Limits (Rationals)

Your argument is correct.If you want a formal proof you can use the definition of continuity in terms of limit of sequences e.g., Define the sequence $x_n=\frac{1}{2^n}$ and $y_n=\frac{\pi}{2^n}$ ...
Marco's user avatar
  • 1,842
2 votes

How to solve this limit without using L'Hopital

By mean value theorem for function $F(t)=\tan(a+h+t)-\tan(a+h)-[\tan(a+t)-\tan a]$, we have $${\tan(a+2h)-\tan(a+h)-[\tan(a+h)-\tan a]=[\tan'(a+h+\theta)-\tan'(a+\theta)]h}$$ for some $|\theta|\leq |h|...
Eric Ley's user avatar
  • 424
2 votes

Without using L'Hopitals rule, how do you get $\lim_{x \to 0} \frac{1 - \cos 5x}{\sin 3x}$?

We have that eventually $$\cos 5x \ge \cos 6x=\frac{1-\tan^2 3x}{1+\tan^2 3x}$$ then $$0\le \frac{1-\cos 5x}{|\sin 3x|}\le \frac{1-\cos 6x}{|\sin 3x|}=\frac{2|\sin 3x|}{(\cos ^23x)(1+\tan^2 3x)} \to \...
user's user avatar
  • 154k
2 votes
Accepted

Without using L'Hopitals rule, how do you get $\lim_{x \to 0} \frac{1 - \cos 5x}{\sin 3x}$?

Note that $$ \frac{1-\cos 5x}{\sin 3x} = \frac{1-\cos 5x}{5x} \cdot \frac{5x}{3x} \cdot \frac{3x}{\sin 3x} $$ Now appeal to known special limits & limit properties.
PrincessEev's user avatar
  • 43.4k
2 votes

Without using L'Hopitals rule, how do you get $\lim_{x \to 0} \frac{1 - \cos 5x}{\sin 3x}$?

@newbie py: Hi welcome to MSE. there are some methods to find this limit. like below $$\lim_{x\to 0}\frac{1-\cos(5x)}{\sin(3x)}\\=\\lim_{x\to 0}\frac{1-\cos(5x)}{\sin(3x)}\times\frac{1+\cos(5x)}{1+\...
Khosrotash's user avatar
  • 24.6k
1 vote
Accepted

A limit of $1^\infty$ form (maybe)

Assuming that the limit exists, let's let $$\begin{align} L &= \lim_{n \to \infty} \left(\frac{(2n)!}{n! n^n} \right)^{1/n} \end{align}$$ Then taking the logarithm, $$\begin{align} \ln(L) &= \...
AlkaKadri's user avatar
  • 2,120
1 vote

$\lim_{x \rightarrow 0} \frac{(x^2 - x^3) ((x + 1)^\sqrt{2} - 1)}{\log(1 + 4 x) (1 - \cos(2 x))}$

$$\frac{(x^2 - x^3) ((x + 1)^\sqrt{2} - 1)}{\log(1 + 4 x) (1 - \cos(2 x))}=\frac{(x+1)^\sqrt2-1}x\cdot\frac{x^2}{1-\cos2x}\cdot\frac{x-x^2}{\log(1+4x)}\;(**)$$ and just as Paramanand told you twice, ...
DonAntonio's user avatar
  • 211k
1 vote

$\lim_{x \rightarrow 0} \frac{(x^2 - x^3) ((x + 1)^\sqrt{2} - 1)}{\log(1 + 4 x) (1 - \cos(2 x))}$

This is one using series expansions for elementary functions $$ \lim_{x \rightarrow 0} \frac{(x^2 - x^3) ((x + 1)^\sqrt{2} - 1)}{\log(1 + 4 x) (1 - \cos(2 x))}=\lim_{x \rightarrow 0} \frac{x^2 ((x + 1)...
RandomGuy's user avatar
  • 1,044
1 vote

How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

Not at all appropriate for a cal one course (or any course), but for the perverse, egregious silliness of it: A circle is parametrized by $\gamma(t) = (\cos t, \sin t)$. Now, geometrically, the ...
peter a g's user avatar
  • 4,923

Only top scored, non community-wiki answers of a minimum length are eligible