Intereting Posts

Closed-form of $\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx$
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Most useful heuristic?
Trace minimization with constraints
Online resources for learning Mathematics
Question about Selberg's formula
pandigital rational approximations to the golden ratio and the base of the natural logarithm
An irreducible $f\in \mathbb{Z}$, whose image in every $(\mathbb{Z}/p\mathbb{Z})$ has a root?
$m$ balls $n$ boxes probability problem
probability, random walk, Markov chain question
Sign of det(UV) in SVD
Comparing average values of an arithmetic function
Show that $\displaystyle{\int_{0}^{\infty}\!\frac{x^{a}}{x(x+1)}~\mathrm{d}x=\frac{\pi}{\sin(\pi a)}}$
How to prove that the roots of this equation are integers?

Past paper Question:

For the following function, determine whether $\lim_{x\to\infty}f(x)$

exists, and compute the limit if it exists. Justify your answers.

$$f(x)= \dfrac{\sin(x)+1}{\left| x \right|}$$

- Is the derivative of an integral always continuous?
- Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$
- Is there a first-order-logic for calculus?
- A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$
- Integrate $\displaystyle \int_{0}^{\pi}{\frac{x\cos{x}}{1+\sin^{2}{x}}dx}$
- How to find area under sines without calculus?

Attempt:Consider the fact that $-1 \le \sin(x) \le 1$ (for all $x$), which implies $0 \le \sin(x) +1\le 2$. Dividing by $\left| x \right|,$

$$\color{green}{

\frac{0}{\left| x \right|}} \le \color{blue}{

\frac{\sin( x)+1}{\left| x \right|}} \le \color{red}{

\frac{2}{\left| x \right|}}$$

Since green tens to $0$, and the red tends to $0$, (via AOL for $\dfrac{1}{x}$ as $x \rightarrow \infty)$, blue will tend to $0$ via the algebra of limits and sandwich theorem, is this correct, or will the absolute value of $x$ effect this?

- Compute $ \sum_{k=1}^{\infty} \text{sech}(2 k)$
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- Eulers proof sum of natural numbers
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- Can we teach calculus without reals?
- Why isn't $f(x) = x\cos\frac{\pi}{x}$ differentiable at $x=0$, and how do we foresee it?
- I want to learn math from zero
- Constants of integration in integration by parts
- Calculating the limit of $^{1/n}$ as $n$ tends to $\infty$
- Find the limit of $\lim_{n\rightarrow\infty}(\frac{1}{2}+\frac{3}{2^2}+…+\frac{2n-1}{2^n})$

Well done.

Note that when $x\to \infty$ implies $|x| \to \infty$

- Infinite number of rationals between any two reals.
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- Integrating the formula for the sum of the first $n$ natural numbers
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- Partial Fractions Expansion of $\tanh(z)/z$
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- Show $\lim\limits_{n\rightarrow\infty}n\left(\frac{1}{n}\sum_{i=1}^{n}f\left(\frac{i}{n}\right)-\int_{0}^{1}f(x)dx\right)=\frac{f(1)-f(0)}{2}$
- “If inaccessible sets exist, their existence is not provable in ZF”