# Questions tagged [calculus]

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

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### Find the limit $\lim \limits_{n\to \infty }\cos \left(\pi\sqrt{n^{2}-n} \right)$

I'd love your help with finding the following limit: $$\lim_{n\to \infty }\cos (\pi\sqrt{n^{2}-n}).$$ I was asked to find this limit, but honestly I believe that it doesn't exist. According to Heine ...
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### $f$ is differentiable and $\lim_{x\to \infty }f(x)=\lim_{x\to -\infty }f(x)=0$

This is quiet a simple question, but still I'm not sure that I am correct. Let $f$ be a differentiable function in $\mathbb{R}$ such that: $\lim_{x\to \infty }f(x)=\lim_{x\to -\infty }f(x)=0.$ We ...
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### Integral with $\sqrt{2x^4 - 2x^2 + 1}$ in the denominator

$$\int\frac{x^{2}-1}{x^{3}\sqrt{2x^{4}-2x^{2}+1}} \: \text{d}x$$ I tried to substitute $x^2=t$ but I am unable to solve it and I also tried to divide numerator and denominator by $x^2$ and do ...
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### Stuck on a homework question: if $t = \frac{1}{x}$

If $\displaystyle t = \frac{1}{x}$ then a) Explain why $\displaystyle\lim_{x \to 0^-}f(x)$ is equivalent to $\displaystyle\lim_{t \to -\infty}f\left(\frac{1}{t}\right)$ b) Using that, ...
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### Approaching from the right or from the left?

Simple question: If I have $\lim_{x \to -3^-}f{(x)}$ and I'm looking at a graph, am I approching -3 from the direction of +$\infty$ to -3 (as in going in the negative direction)? Or am I approaching -...
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### Question on convergence of improper integral

For what values of $\alpha$ is the following integral convergent? $$\int\limits_{-\infty}^{\infty}\frac{|x|^\alpha}{(1+x^2)^m}dx$$ Should the limit comparison theorem be used in this case? I am not ...
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### $\tan(\frac{\pi}{2}) = \infty~$?

Evaluate $\displaystyle \int\nolimits^{\pi}_{0} \frac{dx}{5 + 4\cos{x}}$ by using the substitution $t = \tan{\frac{x}{2}}$ For the question above, by changing variables, the integral can be rewritten ...
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### Golomb sequence

How to compute $n$-th element in golomb's sequence? I've found that $A_n$ is approximately: $$\phi^{2-\phi} * n^{\phi-1},\;\; \text{ where }\;\; \phi \;\;\text {is golden ratio}.$$ As far as I ...
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### Bounding ${(2d-1)n-1\choose n-1}$

Claim: ${3n-1\choose n-1}\le 6.25^n$. Why? Can the proof be extended to obtain a bound on ${(2d-1)n-1\choose n-1}$, with the bound being $f(d)^n$ for some function $f$? (These numbers describe ...
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### Eigenfunction of $(a(x) f^{II})^{II}= - \lambda^2f$

I need the eigenfunctions $f$ and eigenvalues $\lambda$ of $(a(x) f^{II}(x))^{II}= - \lambda^2f$ for a given $a(x)$. For $a(x)$ constant the solution is a combination of sin, cos, sinh and cosh. ...
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### First order logic and higher order logics?

I hear that Prolog is based in first-order logic. This makes me wonder, C/C++ are based on which higher order logics? If this question is incorrect, please point out that. and how are these logics ...
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### Bézier Curve and Gravitational Pull

Okay, so I'm trying to manipulate an object programmatically. When it gets near another object, lets say a globe. I want the globe to have a gravitational pull on the original object. This isn't so ...
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### Need help proving the following relationship

$$\frac{\partial \phi}{\partial \xi} ( \parallel \mathbf{x} - \xi_i \parallel) = - \frac{\partial \phi}{\partial x} ( \parallel \mathbf{x} - \xi_i \parallel)$$ from page 14 of these lecture ...
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### Evaluating $\int_{0}^{1} \frac{dx}{1+{}_2F_{1}\left(\frac{1}{n},x;\frac{1}{n};\frac{1}{n}\right)}$

On a lark (as a followup to this question), I was playing around with Wolfram alpha, and it seems that \int_{0}^{1} \frac{dx}{1+{}_2F_{1}\left(\frac{1}{n},x;\frac{1}{n};\frac{1}{n}\right)} = \frac{\...
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### Implicit differentiation misunderstanding

I'm trying to see why my textbook's solution is correct and mine isn't. "Find an expression in terms of $x$ and $y$ for $\displaystyle \frac{dy}{dx}$, given that $x^2+6x-8y+5y^2=13$ First, the ...
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### Does this equality always hold?

Is it true in general that $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x} \int_0^{x} f(u,x) \mathrm{d}u = \int_0^{x} \left( \frac{\mathrm{d}}{\mathrm{d}x} f(u,x) \right)\mathrm{d}u +f(x,x )$ ? Thank ...
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### the approximation of $\log(266)$?

Consider the following exercise: Of the following, which is the best approximation of $\sqrt{1.5}(266)^{1.5}$? A 1,000 B 2,700 C 3,200 D 4,100 E 5,300 The direct idea is using the "...
### How to convince a layperson that the $\pi = 4$ proof is wrong?
The infamous "$\pi = 4$" proof was already discussed here: Is value of $\pi = 4$? And I have read all the answers, yet I think that they will not be of much help to me if I try to explain this thing ...
Let $p$ be a function of the form $\mathbb{R}^2 \to \mathbb{R}$. How do i find the derivatives of the following expressions with respect to p(x,y) : a. $\int_\mathbb{R^2} p$ b. \$ \dfrac{\partial p}{...