0
votes
1answer
26 views

Different ways to formally define trigonometric functions

When I first learnt trigonometric functions I was in highschool and obviously the explanation they gave me was mostly intuitive. Now that I have taken my first curse of calculus I learnt a formal ...
11
votes
3answers
358 views

A doubt in the rigorous definition of limits.

I studied the definition of limits today, and I think I mostly understood it, but I have a little doubt. In the definition: $f(x)$ is defined on some open interval containing $a$, except at possibly ...
0
votes
2answers
49 views

Limits to infinity?

As a part of homework, I was asked What does $\lim_{x\to a} f(x)=\infty$ mean? In an earlier calculus class I was taught that in order for $L=\lim_{x\to a}f(x)$ to exist, we need that ...
0
votes
2answers
78 views

How does the epsilon-delta definition define a limit?

I understand what the epsilon-delta definition is saying in regards to the distance from a point c and the distance from your limit, but I don't understand how this defines a limit. Any help is ...
1
vote
4answers
59 views

Vertical bar notation: $\frac{d}{dt}|_{t=0}f(a+tv)=$?

The definition of the directional derivative is $D_vf(a)$=$\left.\dfrac{d}{dt}\right|_{t=0}f(a+tv)=\lim \limits_{t\to0}\dfrac{f(a+tv)-f(a)}{t}$. However, I do not understand how this bar notation is ...
0
votes
1answer
45 views

Fiding a derivative

I need to find the derivative of $\sqrt{x^2+3x}$ using the definition of derivative. e.g. $\frac{f(x)-f(a)}{x-a}$ as x->a. Normally I get these but the $x^2$ is messing me up. I am at $$\lim ...
0
votes
2answers
61 views

I Need Help Understanding the Formal Definition of A Limit

I had been taught the formal definition of a limit with quantifiers. For me, it is very hard to follow and I understand very little of it. I was told that: $$\text{If} \ \lim\limits_{x\to a}f(x)=L, \ ...
0
votes
0answers
26 views

Definition of $\partial^2{u}/ \partial {n^2}$

I want to solve a boundary value problem on a square, that one of the boundary condition is $$\frac{\partial^2{u}}{ \partial {n^2}}=0.$$ The test example that I want to solve is: $$\Delta^2u=f,$$ ...
17
votes
5answers
476 views

$\epsilon, \delta$…So what?

Over the course of my studies I often encounter phrases in reference material of the type "and this avoids the need for using $\epsilon$, $\delta$ definitions" or "by this we can omit those ...
3
votes
2answers
93 views

Radius of Convergence and its application to a Power Series including $x^{2n}$ rather than $x^n$

(Radius of Convergence) Consider the Power Series $f(x)=\sum_{n=0}^{+ \infty}a_n x^n$, the radius of convergence $\rho$ can be found using $$\rho = \displaystyle \lim_{n \to + \infty} \left| ...
0
votes
1answer
32 views

Question on definition of little o

I would like to generalize the definition of little o. The definition from Wikipedia is as such: Let $f$ and $g$ be two real valued functions. We write $f(x) = o(g(x))$ as $x \to \infty$ if for all ...
0
votes
1answer
56 views

Difference between the concepts of graph and trace

I'm a little confused with the definition of graph and trace. If I have a function (or a curve) $f:\mathbb R\to \mathbb R,\ f(t)=t^2$ and I draw the graph we have a parabola since the graph is the ...
0
votes
1answer
52 views

Definition of continuity without limit

Without limits, I define $f(x)$ is continuous at $x=a$, when it: $f(a)$ exists; For every $d>0$, in the close interval $[a-d,a+d]$, there exist a maximum $M$ and a minimum $m$; For every ...
1
vote
2answers
70 views

Proof that the derivative of a linear function is $0$.

We have a function $f(x)=x$ which is defined and is continuous on the set $S$ of all real numbers. The derivative at point $x$, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}h$$ Using the theorems of ...
0
votes
3answers
121 views

Solution of a differential equation having a singularity (not everywhere defined) [closed]

Remind me about ordinary differential equations, whose solutions are not everywhere defined (have a singularity). I want to remember the exact definition of a solution with singularity, which I ...
0
votes
1answer
66 views

Limit of Identity Function vs. limit of Squaring Function

$$\lim_{x\rightarrow a} x = a$$ and $$\lim_{x\rightarrow a} x^2 = a^2$$ $f(x)=x^2=x \times x$, i.e.: two identity functions. I'm a bit confused on how $x^2$ can be interpreted as being ...
6
votes
5answers
148 views

The derivative of $e^x$ using the definition of derivative as a limit and the definition $e^x = \lim_{n\to\infty}(1+x/n)^n$, without L'Hôpital's rule

Let's define $$ e^x := \lim_{n\to\infty}\left(1+\frac{x} {n}\right)^n, \forall x\in\Bbb R $$ and $$ \frac{d} {dx} f(x) := \lim_{\Delta x\to0} \frac{f(x+\Delta x) - f(x)} {\Delta x} $$ Prove that ...
1
vote
2answers
82 views

What *really* are the local maxima and local minima

In math is the local max and local min just any peak ... point where slope of the function changes from positive to negative or vice-versa... Or are the LOCAL max and min just the highest point of the ...
0
votes
1answer
199 views

Why is the direct substitution property so specific

Mt text book states the Direct Substitution Property as If f is a polynomial or a rational function and a is in the domain, then $$\begin{align*} \lim_{x\to a} f(x)=f(a) \end{align*}$$ Why does ...
5
votes
2answers
96 views

Differential — Mathematically conform?

In calculus, I know that one defined the differential quotient $$\frac{dy}{dx} := \lim\limits_{h \rightarrow 0}{\frac{y(x+h)-y(x)}{h}}$$ I learned that it is not a quotient, but can be treated as one ...
2
votes
2answers
78 views

Is it true that $ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $?

Is the equality below true? $$ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $$
4
votes
4answers
205 views

Using the definition of the derivative prove that if $f(x)=x^\frac{4}{3}$ then $f'(x)=\frac{4x^\frac{1}{3}}{3}$

So I have that $f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ and know that applying $f(x)=x^\frac{4}{3}=f\frac{(x+h)^\frac{4}{3} -x^\frac{4}{3}}{h}$ but am at a loss when trying to expand ...
3
votes
4answers
739 views

Continuity on open interval

A function is said to be continuous on an open interval if and only if it is continuous at every point in this interval. But an open interval $(a,b)$ doesn't contain $a$ and $b$, so we never ...
2
votes
0answers
852 views

Continuity on open and closed intervals

I will be taking Calculus I soon, and I just want to make sure I understand some concepts correctly. So far, reading my book for Calculus I, I've encountered the definition of continuity as being ...
3
votes
3answers
126 views

Calculating derivative by definition vs not by definition

I'm not entirely sure I understand when I need to calculate a derivative using the definition and when I can do it normally. The following two examples confused me: $$ g(x) = \begin{cases} x^2\cdot ...
2
votes
2answers
159 views

Defining infinitesimals

Can such definition of infinitesimals hold? $$\mathrm{d} x :=a:(a>0 \;\And\; \forall b \in \mathbb{R}^+\backslash \{ a \}\;(a<b))$$ And, if the above definiton works, then obviously ...
0
votes
1answer
77 views

question about epsilon, delta limit definition

Sometimes, when describing the closeness of $x$ to $a$ as being less than $\delta$, it's stated as $|x-a|<\delta$ and sometimes it's stated as $0<|x-a|<\delta$. What is the " $0<$ " part ...
1
vote
3answers
127 views

For $\lim_{x \rightarrow a}f(x) = L$, if $f(x)$ is not a constant, is $\delta(\epsilon)$ always a monotonically increasing function?

Alternatively, how does the definition of a limit guarantee that if $f(x)$ is not a constant, then a small $\epsilon$ will give me a small $\delta$?
2
votes
1answer
571 views

Definition of local maxima, local minima

Wikipedia says that: A real-valued function f defined on a real line is said to have a local (or relative) maximum point at the point x∗, if there exists some ε > 0 such that f(x∗) ≥ f(x) when |x − ...
2
votes
3answers
983 views

What's the difference between direction, sense, and orientation?

I'm trying to understand the difference between the sense, orientation, and direction of a vector. According to this, sense is specified by two points on a line parallel to a vector. Orientation is ...
1
vote
1answer
30 views

Change Along A Tangent Line

I am taking some time to review differentials. What I don't quite get is why the change along the tangent line is $f'(x) \Delta x$, and how it leads to $f'(x)dx$
1
vote
1answer
194 views

complement of a function $f: \{2n | n\in \mathbb{N}_0 \}: n \rightarrow n+1$

i am reading a textbook here and i saw, there is notion of Complement of a function. or Negation of a function definiton, this is whow i understood but it is definitely wrong how i do it, i know. in ...
3
votes
2answers
165 views

Motivation for a new definition of the derivative using the concept of average velocity

As everyone knows, for a function $ f: \mathbb{R} \to \mathbb{R} $ and a point $ a \in \mathbb{R} $, we say that the derivative of $ f $ at $ a $ equals $ L $ if and only if $$ \lim_{x \to a} ...
2
votes
4answers
521 views

Definition for Covariant Derivative

What is simple definition of the covariant derivative that looks like the definition of the derivative of a function in calculus? definition of the derivative of a function in calculus is: $$\frac ...
10
votes
3answers
324 views

Why does the definition of limits of a function have strict inequality?

Definition (As written in Michael Spivak's Calculus) The function $f$ approaches a limit $l$ near $a$ means: for every $\epsilon >0$ there is some $\delta > 0$ such that, for all $x$, if ...
2
votes
2answers
170 views

Defining $0^0=1$

I've read in several places that defining $0^0=1$ is convenient in several (primarily discrete) settings. One argument on Wikipedia in favor of this definition was the need of a special case for the ...
1
vote
1answer
234 views

$\epsilon - \delta$ definition of a limit

Where can I find a good explanation of the $\epsilon - \delta$ definition of a limit. I have tried looking at my textbook and it doesn't make much sense, and I have also looked on Google as well ...
14
votes
4answers
551 views

What's wrong with this “backwards” definition of limit?

Can anyone please give an example of why the following definition of $\displaystyle{\lim_{x \to a} f(x) =L}$ is NOT correct?: $\forall$ $\delta >0$ $\exists$ $\epsilon>0$ such that if ...
2
votes
2answers
126 views

Motivation behind definitions of the Integral without reference to Derivatives

If a (definite) Integral can simply be calculated as the difference of two Antiderivatives and Antiderivatives are simply the "reverse process" of Differentiation. Then it seems to me that the ...
7
votes
1answer
446 views

Why $\lim\limits_{n\to \infty}\left(1+\frac{1}{n}\right)^n$ doesn't evaluate to 1?

I am trying to identify what the flaw is exactly when reasoning about a limit such as the definition of $\mathbf e$: $$ \lim_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^n={e} $$ Now, I know ...
4
votes
2answers
130 views

Defining a Differentiable Function in a Non-Pointwise Manner

Is there a way to define a differentiable function in a non-pointwise manner? That is without defining function differentiable at a point first. Just like we define a continuous function through open ...
1
vote
3answers
2k views

why do we use 'non-increasing' instead of decreasing?

In english based math language it seems that non-increasing $\Longleftrightarrow$ less or equal (non-strict decreasing) decreasing $\Longleftrightarrow$ strict less ( strict decreasing) ...
1
vote
3answers
97 views

Oddities in the Definition of IntegralCosinus ${\rm Ci}(x)$

Reading the defintion of the IntegralCosinus $$ {\rm Ci}(x) = \gamma + \ln x + \int_0^x\frac{\cos t-1}{t}\,dt $$ I wonder what happens, if I to split the function in the integral: $$ ...
0
votes
2answers
492 views

How many 'supremum(s)' and 'infimum(s)' can a set have?

I am learning calculus/real analysis with Apostol's Calculus (2nd Edition). I have a doubt about the grammer of this book. Apostol, everywhere, uses a supremum (or a least upper bound) and an infimum ...
2
votes
0answers
209 views

How to prove that $e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)} = \lim_{n \to \infty} \sqrt[n]{n\#} $?

While reading this post, I stumbled across these definitions (Wiki_german) $$e = \lim_{n \to \infty} \sqrt[n]{n\#}$$ and $$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}.$$ The last one seems ...
5
votes
5answers
432 views

Can series start with $-∞$?

So, I decided to dig a little deeper into numerical integration because we hardly had any of that in my analysis class. I've come across this method for improper integrals: Метод Самокиша (not in ...
1
vote
3answers
181 views

After saw this piece of discussion, i ask myself what is the most rigorous definition of the circle, but i can't figure it out

Discussion: Is value of $\pi$ = 4? so what is the "real definition" of a circle? i think the original solution from wikipedia is too ambigous, i couldn't find why the circumference of the circle is ...
16
votes
5answers
1k views

Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
20
votes
3answers
1k views

Definition of convergence of a nested radical $\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4+\cdots}}}}$?

In my answer to the recent question Nested Square Roots, @GEdgar correctly raised the issue that the proof is incomplete unless I show that the intermediate expressions do converge to a (finite) ...
5
votes
1answer
148 views

What motivates discrepancies between the definitions of “continuous” and “limit”?

I am working from Munkres' Analysis, and I've converted his definitions slightly to make them easier to compare. In the table below, you can fill in the blanks in the top row with words from the ...