4
votes
4answers
215 views

Prove that these two curves have the same length

My midterms are approaching, and I was going through some of our past Calculus midterms when I stumbled upon this question from 1996: Show that these two curves, $$(\Gamma) : \frac ...
3
votes
0answers
50 views

How to give an epsilon-delta proof of this limit statement? [duplicate]

Although I know a couple of proofs of the statement $$ \lim_{x \to 0 } \frac{\sin x}{x} = 1, $$ I would like to be able to come up with a proof using the definition of the limit (i.e. an ...
0
votes
0answers
29 views

Extreme Value Theorem: from 1 variable to several variables?

When studying functions of several variables, one useful idea is to use the same result for one variable. For example, to prove that a local extremum of a function $f(x_1,\dots,x_n)$ is a critical ...
2
votes
1answer
42 views

If $|\nabla F| > 1$ and $|F| \le 1$, is there a zero nearby?

I saw this claim, stated without much explanation, in an article I'm reading: Let $F:\mathbb{R}^n\to\mathbb{R}$ be a $C^1$ function which satisfies $|\nabla F|>1$ everywhere. We know that ...
2
votes
2answers
69 views

Proving the derivative is $0$ at the extremum and all derivatives are $0$.

The pictures below show the proof that Apostol uses in his book. I can't understand why Apostol introduces the function $Q(x)$ and proves the theorem by contradiction using the sign preserving ...
1
vote
2answers
217 views

Using inequalities and limits

Is it possible to say: $$ If \ f(x) \ and \ g(x) \ both \ have \ limits \ as \ x\to p\ and \ f(x) \le g(x), \ then \lim_{x \to p} f(x)\le \lim_{x \to p} g(x). $$ My proof(Edit: Proof is wrong due to ...
1
vote
2answers
212 views

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 $$\lim_{x \to a}[f(x)+g(x)]=A+B$$ ...
0
votes
7answers
256 views

Why Not Define $0/0$ To Be $0$?

For every number $x$, $x\times 0=0$, hence $\dfrac{0}{0}$ can be any number! So $\dfrac{0}{0}$ "is knows as indeterminate" [1]. But what if we define it to be $0$? I already have an answer, but ...
1
vote
4answers
135 views

Elegant or elementary evaluation of $\lim\limits_{x\to 0} \left( \frac{1}{x}-\frac{1}{\sin(x)} \right) $ [duplicate]

I give math tutoring and was wondering about the following limit. I found the answer but I was wondering if someone has a nicer explanation than the one I am giving where I use L'Hôpital's rule twice. ...
5
votes
2answers
266 views

Is my proof correct about limit of $\sin\left(\frac{1}{x}\right)$?

Apostol's book Calculus asks to show that there is not a value $A$ such that $f(x)=\sin\left(\frac{1}{x}\right)\to A$ when $x \to 0$. And my proof is: Suppose for the sake of contradiction that there ...
6
votes
11answers
778 views

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

How to prove that $$\lim\limits_{x\to0}\frac{\tan x}x=1?$$ I'm looking for a method beside L'Hospital's rule.
1
vote
1answer
101 views

Identity concerning complete elliptic integrals

It can be easily checked that both the complete elliptic integrals $K(k), K'(k)$ satisfy the same second order differential equation $$kk'^{2}\frac{d^{2}y}{dk^{2}} + (1 - 3k^{2})\frac{dy}{dk} - ky = ...
16
votes
5answers
473 views

A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$

This is a problem from "A Course of Pure Mathematics" by G H Hardy. Find the limit $$\lim_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$$ I had solved it long back (solution presented in my blog ...
1
vote
4answers
2k views

Prove that a continuous function on a closed interval attains a maximum

As the title indicates, I'd like to prove the following: If $f:\mathbb R\to\mathbb R$ is a continuous function on $[a,b]$, then $f$ attains its maximum. Now, I do have a working proof: $[a,b]$ ...
7
votes
11answers
807 views

limit question: $\lim\limits_{n\to \infty } \frac{n}{2^n}=0$

$$ \lim_{n\to\infty}\frac n{2^n}=0. $$ I know how to prove it by using the trick, $2^n=(1+1)^n=1+n+\frac{n(n-1)}{2}+\text{...}$ But how to prove it without using this?
3
votes
1answer
85 views

Prove: if $f(x) =x\sin (\pi x)$ then $f'(x)$ vanishes at a unique point in $ ( n + 1/2, n) $

Let $ f(x) = x\sin (\pi x), x > 0 $. Then prove that for all natural numbers n, $f'(x)$ vanishes at a unique point in $ ( n + 1/2, n) $ The given solution shows a graph, but is there any algebraic ...
5
votes
5answers
540 views

Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction [duplicate]

I was looking at: $$\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$$ It's pretty easy proving the above using induction, but I was wondering what is the actual way of getting this equation?
9
votes
3answers
352 views

Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?

In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one? I can of course ...
4
votes
2answers
1k views

How is Leibniz's rule for the derivative of a product related to the binomial formula? [duplicate]

Possible Duplicate: “Binomial theorem”-like identities The binomial formula describes the expansion of the $n$th power of the sum $(a+b)$: $$(a+b)^n = \sum_{k = 0}^n {n\choose ...
3
votes
1answer
348 views

proof of l'Hôpital's rule that minimizes special-casing

A simple form of l'Hôpital's rule looks like this: If $u$ and $v$ are functions with $u(0)=0$ and $v(0)=0$, the derivatives $\dot{v}(0)$ and $\dot{v}(0)$ are defined, and the derivative $\dot{v}(0)\ne ...