Skip to main content

Questions tagged [calculus]

For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.

Filter by
Sorted by
Tagged with
554 votes
30 answers
232k views

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

How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my math ...
FUZxxl's user avatar
  • 9,317
1296 votes
27 answers
145k views

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...
BBSysDyn's user avatar
  • 16.2k
127 votes
3 answers
10k views

Are all limits solvable without L'Hôpital Rule or Series Expansion

Is it always possible to find the limit of a function without using L'Hôpital Rule or Series Expansion? For example, $$\lim_{x\to0}\frac{\tan x-x}{x^3}$$ $$\lim_{x\to0}\frac{\sin x-x}{x^3}$$ $$\...
lab bhattacharjee's user avatar
59 votes
14 answers
24k views

How to prove that $\log(x)<x$ when $x>1$? [duplicate]

It's very basic but I'm having trouble to find a way to prove this inequality $\log(x)<x$ when $x>1$ ($\log(x)$ is the natural logarithm) I can think about the two graphs but I can't find ...
Gianolepo's user avatar
  • 2,507
86 votes
5 answers
23k views

Limit of the nested radical $x_{n+1} = \sqrt{c+x_n}$

(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12) For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x > 0$. Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, ...
cnuulhu's user avatar
  • 1,005
274 votes
32 answers
134k views

Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?

A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int\limits_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$ Well, can ...
user avatar
358 votes
7 answers
52k views

How can you prove that a function has no closed form integral?

In the past, I've come across statements along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations: addition/...
Simon Nickerson's user avatar
218 votes
21 answers
159k views

Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$

How to prove $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
Jichao's user avatar
  • 8,018
110 votes
15 answers
100k views

How to prove that exponential grows faster than polynomial?

In other words, how to prove: For all real constants $a$ and $b$ such that $a > 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$ I know the definition of limit but I feel that it's not enough to ...
LeafGlowPath's user avatar
  • 7,243
265 votes
9 answers
35k views

Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$

I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$, which is a pretty interesting and nice ...
user 1591719's user avatar
  • 44.3k
21 votes
3 answers
3k views

Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$. [duplicate]

Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, \mathrm dx.$$ How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead ...
StationaryTraveller's user avatar
112 votes
11 answers
15k views

Finding the limit of $\frac {n}{\sqrt[n]{n!}}$

I'm trying to find $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$ I tried couple of methods: Stolz, Squeeze, D'Alambert Thanks! Edit: I can't use Stirling.
32 votes
3 answers
18k views

Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means [duplicate]

Prove that if $\lim_{n \to \infty}z_{n}=A$ then: $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n}...
Mykolas's user avatar
  • 1,357
12 votes
4 answers
1k views

Analyzing limits problem Calculus (tell me where I'm wrong).

I came accross: $$\lim_{x \to\ 0}\frac{x\cos x - \log (1 + x)}{x^{2}}$$ I tried the following please tell me where I 'm wrong: $$\lim_{x \to\ 0}\frac{x\cos x - \log (1 + x)}{x^{2}} $$ $$\text{(...
mnulb's user avatar
  • 3,381
78 votes
5 answers
10k views

What am I doing when I separate the variables of a differential equation?

I see an equation like this: $$y\frac{\textrm{d}y}{\textrm{d}x} = e^x$$ and solve it by "separating variables" like this: $$y\textrm{d}y = e^x\textrm{d}x$$ $$\int y\textrm{d}y = \int e^x\textrm{d}x$...
Mark Eichenlaub's user avatar
40 votes
2 answers
10k views

Fourier series of Log sine and Log cos

I saw the two identities $$ -\log(\sin(x))=\sum_{k=1}^\infty\frac{\cos(2kx)}{k}+\log(2) $$ and $$ -\log(\cos(x))=\sum_{k=1}^\infty(-1)^k\frac{\cos(2kx)}{k}+\log(2) $$ here: twist on classic log of ...
Peder's user avatar
  • 2,192
164 votes
26 answers
38k views

Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?

Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very slowly?...
bryn's user avatar
  • 9,764
120 votes
10 answers
15k views

Proof of Frullani's theorem

How can I prove the Theorem of Frullani? I did not even know all the hypothesis that $f$ must satisfy, but I think that this are Let $\,f:\left[ {0,\infty } \right) \to \mathbb R$ be a a continuously ...
August's user avatar
  • 3,543
65 votes
6 answers
6k views

Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$

This is being asked in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions, and here: List of abstract duplicates. What methods can be used to evaluate the limit $$...
Eric Naslund's user avatar
  • 72.3k
52 votes
4 answers
6k views

Universal Chord Theorem

Let $f \in C[0,1]$ and $f(0)=f(1)$. How do we prove $\exists a \in [0,1/2]$ such that $f(a)=f(a+1/2)$? In fact, for every positive integer $n$, there is some $a$, such that $f(a) = f(a+\frac{1}{n})$...
Ma.H's user avatar
  • 1,043
45 votes
8 answers
5k views

How to prove $\sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6$?

I'd like to find out why \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6 \end{align} I tried to rewrite it into a geometric series \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = \sum_{n=...
TheWaveLad's user avatar
  • 1,495
22 votes
1 answer
5k views

Getting different answers when integrating using different techniques

Question: Is it possible to get multiple correct results when evaluating an indefinite integral? If I use two different techniques to evaluate an integral, and I get two different answers, have I ...
Xander Henderson's user avatar
  • 29.8k
19 votes
1 answer
3k views

The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ as $n\rightarrow\infty$ [closed]

How to compute this limit: $$\lim_{n\to\infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\cdots+\frac{n}{n^2+n^2}\right)$$ Please give me some hint.
Barbara's user avatar
  • 351
193 votes
12 answers
92k views

What is $dx$ in integration?

When I was at school and learning integration in maths class at A Level my teacher wrote things like this on the board. $$\int f(x)\, dx$$ When he came to explain the meaning of the $dx$, he told us ...
Sachin Kainth's user avatar
73 votes
11 answers
26k views

Where is the flaw in this "proof" that 1=2? (Derivative of repeated addition)

Consider the following: $1 = 1^2$ $2 + 2 = 2^2$ $3 + 3 + 3 = 3^2$ Therefore, $\underbrace{x + x + x + \ldots + x}_{x \textrm{ times}}= x^2$ Take the derivative of lhs and rhs and we get: $\...
user avatar
118 votes
13 answers
17k views

Calculating the integral $\int_0^\infty \frac{\cos x}{1+x^2}\, \mathrm{d}x$ without using complex analysis

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos x}{1+x^2}\,\mathrm{d}x$$
Martin Gales's user avatar
  • 6,878
176 votes
9 answers
34k views

Why is $1^{\infty}$ considered to be an indeterminate form

From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are ...
user avatar
165 votes
16 answers
16k views

Why does factoring eliminate a hole in the limit?

$$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5} = \lim_{x\rightarrow 5} (x+5)$$ I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and ...
Emi Matro's user avatar
  • 5,033
11 votes
9 answers
6k views

How to compute the formula $\sum \limits_{r=1}^d r \cdot 2^r$?

Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$ how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$ Thank you
q0987's user avatar
  • 223
144 votes
16 answers
11k views

Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$

Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums. ...
Mike Spivey's user avatar
  • 55.7k
132 votes
9 answers
27k views

Prove that $C e^x$ is the only set of functions for which $f(x) = f'(x)$

I was wondering on the following and I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $ e^x$ is the same as the function itself....
Timo Willemsen's user avatar
70 votes
8 answers
141k views

Finding $\int x^xdx$

I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this: Let $u=x^x$. $$\begin{align} \int x^x \, dx&=\int u \, du\\[6pt] &=\frac{u^2}{2}\\[6pt] &=\dfrac{\left(x^...
Garmen1778's user avatar
  • 2,338
46 votes
11 answers
23k views

Computing the integral of $\log(\sin x)$

How to compute the following integral? $$\int\log(\sin x)\,dx$$ Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we find ...
Kolya's user avatar
  • 505
93 votes
15 answers
31k views

Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$

I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$ What do you think? It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} \mathrm ...
TCL's user avatar
  • 14.3k
56 votes
3 answers
3k views

Evaluating $\int P(\sin x, \cos x) \text{d}x$

Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$. For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$. Is there a general method which allows us to evaluate the ...
25 votes
5 answers
36k views

How to show that $f(x)=x^2$ is continuous at $x=1$? [closed]

How to show that $f(x)=x^2$ is continuous at $x=1$?
user avatar
45 votes
4 answers
6k views

Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $

Is there any way to show that $$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^k}\left( {\...
Pedro's user avatar
  • 122k
29 votes
6 answers
7k views

How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?

I can show that the following limit exists but I am having difficulties to find it. It is $$\lim_{n\to \infty} \sum_{k=1}^n \frac{k^n}{n^n}$$ Can someone please help me?
martin's user avatar
  • 931
127 votes
8 answers
85k views

Why is the derivative of a circle's area its perimeter (and similarly for spheres)?

When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle. Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is ...
bryn's user avatar
  • 9,764
172 votes
8 answers
43k views

Lesser-known integration tricks

I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now ...
58 votes
6 answers
61k views

Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist

I've been trying to solve the following problem: Suppose that $f$ and $f'$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\...
saurs's user avatar
  • 1,387
164 votes
1 answer
22k views

How to determine with certainty that a function has no elementary antiderivative?

Given an expression such as $f(x) = x^x$, is it possible to provide a thorough and rigorous proof that there is no function $F(x)$ (expressible in terms of known algebraic and transcendental functions)...
hesson's user avatar
  • 2,074
106 votes
8 answers
48k views

Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$

Compute $$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
user 1591719's user avatar
  • 44.3k
17 votes
8 answers
15k views

What are the recommended textbooks for introductory calculus?

I've already taken my calculus sequence and I'm interested in brushing up and staying sharp on the basics. So far, my calculus background is limited to single-variable calculus, which I applied in my ...
61 votes
3 answers
6k views

Areas versus volumes of revolution: why does the area require approximation by a cone?

Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$. Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder. ...
Eric O. Korman's user avatar
34 votes
7 answers
17k views

Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$

For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$. My attempt: Let $$\begin{align*} f_n(x) &= \frac{\ln\left(1-2 \left(a+\frac{1}...
Stephen Dedalus's user avatar
39 votes
12 answers
89k views

Why $\sum_{k=1}^{\infty} \frac{k}{2^k} = 2$? [duplicate]

Can you please explain why $$ \sum_{k=1}^{\infty} \dfrac{k}{2^k} = \dfrac{1}{2} +\dfrac{ 2}{4} + \dfrac{3}{8}+ \dfrac{4}{16} +\dfrac{5}{32} + \dots = 2 $$ I know $1 + 2 + 3 + ... + n = \dfrac{n(n+1)}{...
jeebee's user avatar
  • 433
123 votes
18 answers
53k views

Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{...
Mike Spivey's user avatar
  • 55.7k
57 votes
6 answers
14k views

Formal basis for variable substitution in limits

So we do a lot of variable substitution in limits at school. Stuff like $\lim\limits_{x\to5}\ (x+1)^2\ =\ \lim\limits_{y\to6}\ y^2$, where we define the substitution $y = x + 1$. But I've never been ...
Jack M's user avatar
  • 27.9k
39 votes
4 answers
26k views

Finding a limit using change of variable- how come it works? [duplicate]

I'm a student just starting calculus in college, and my math skills are pretty stale. So... how come finding limits using change of variable works? For example: $$\lim_{x \to 1}\frac{x\cos(x-1) -1}{...
AK_'s user avatar
  • 571

1
2 3 4 5
270