Articles of limits

Prove that $\{x_1=1,\,x_{n+1}=\frac {x_n}2+\frac 1{x_n} \}$ converges when $n \to \infty$

I want to prove that the sequence defined by $\{x_1=1,\,x_{n+1}=\frac {x_n}2+\frac 1{x_n} \}$ has a limit. By evaluating the sequence I notice that the sequence is strictly monotonically decreasing starting from $x_2=1.5$. It seems to suggest itself to prove that the sequenced is bounded by $1\le\big( x_n \big)_{n\ge1} \le 1.5$ and to prove that it […]

Extended $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ limit law?

So I’ve learned that $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ is true and the following picture really helped me get an intuitive feel for why that is I have been told that this limit is true whenever the argument of sine matches the denominator and they both tend to zero. That is, $$\lim_{x \rightarrow 0}{\frac{\sin(5x)}{5x}} = […]

How to answer the question from Calculus by Michael Spivak Chapter 5 Problem 14

Prove that if $\lim\limits_{x\rightarrow0}{\frac{f(x)}x}=l$ and $b\neq 0$, then $\lim\limits_{x\rightarrow0}{\frac{f(bx)}x}=bl$. Hint: Write $\frac{f(bx)}x=b\frac{f(bx)}{bx}$ What happens if $b=0$? Part 1 enables us to find $\lim\limits_{x\rightarrow0}{\frac{\sin{2x}}{x}}$ in terms of $\lim\limits_{x\to0}\frac{\sin x}x$. Find this limit in another way. This is a question from Calculus by Michael Spivak Chapter 5 Problem 14.

Asymptotic solving of a hyperbolic equation

The solition and anti-solition nonlinear equation is given as: My problem is that, how do we get the next equation after considering asyptotic behaviour? Resource: (solition) at page 38

How to evaluate $\lim\limits_{n\to +\infty}\frac{1}{n^2}\sum\limits_{i=1}^n \log \binom{n}{i} $

I have to compute: $$ \lim_{n\to +\infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\binom{n}{i}.$$ I have tried this problem and hit the wall.

Find $\lim_{x\rightarrow 0} \frac{\cos x – 1}{x}$

I’m trying to find the following limit: $$\lim_{x\rightarrow 0} \frac{\cos x – 1}{x}$$ I tried to use squeeze theorem but it’s not making much sense. I did the following: $$\begin{align} &\lim_{x\rightarrow 0} \frac{\cos x – 1}{x} \\ -1 \le &\lim_{x\rightarrow 0} (\cos x)(x^{-1}) \le 1 \\ \lim_{x\rightarrow 0} -x^{-1} \le &\lim_{x\rightarrow 0} \cos x \le […]

An unknown limit with nth root: $\lim\limits_{n\to\infty}n(x^{1/n}-1)$

This question already has an answer here: How to find $\lim\limits_{n\to\infty} n·(\sqrt[n]{a}-1)$? 4 answers

Proving the limits of the sum of two functions is equal to the sum of the limits

I am new to proving in math so I want to know if this informal proof of limits is possible: Theorem: If $\lim_{x \to a}f(x)=A$ and $\lim_{x \to a}g(x) = B$, then $$\lim_{x \to a}[f(x)+g(x)]=A+B$$ $\lim_{x \to a}[f(x) + g(x)] = A + B$ is the same as $\lim_{x\to a}[(f(x)-A) + (g(x) – B)]=0$. Also, […]

How do I solve this limit: $\lim _{x \to 0} \left(\frac{ \sin x}{x}\right)^{1/x}$?

$$\lim _{x \rightarrow 0} \left(\frac{ \sin x}{x}\right)^{1/x}$$ I have spent an hour on the above limit and have no work to show. I tried using L’Hopital’s Rule, but just kept going around in circles. Any help would be appreciated. Thank you.

Rigorous Definition of One-Sided Limits

In a typical first-year Calculus course professors typically tend to put a lot of emphasis on making visual connections when working with “one-sided” limits or derivatives. This is something I find particularly distressing as I prefer to think as abstractly as possible, and most professors’ emphasis on making visual connections comes at a price. Professors […]