# Tagged Questions

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

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### If a Taylor series about $x_{0}$ has a radius of converges $R = \infty$

If a Taylor series about $x_{0}$ has a radius of converges $R = \infty$, what does it say about the Taylor remainder about $x_{0}$? Does it say anything about the Taylor polynomial about $x_{0}$? ...
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### Continuous functions and limits inequalities

Is it true that if $f(x)$ and $g(x)$ are both continuous functions and $f(x) \leq g(x)$ for all $x \in \mathbb{R}$, then $$\lim_{x\to\infty}f(x) \leq \lim_{x\to\infty}g(x)?$$ It makes sense to me ...
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### There exists $c \in [0,1]$ for which $\int_{0}^{1}\sin(x^3) = \int_{0}^{c}\sin(x^2)$

T/F: There exists $c \in [0,1]$ for which $\int_{0}^{1}\sin(x^3) = \int_{0}^{c}\sin(x^2)$ I know the answer it true, and I already saw the proof. What I don't get is this: $\sin$ is monotonically ...
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### $\lim_{x\to \infty} \ln x=\infty$

I'm reading the following reasoning: Since $\underset{n\to \infty}{\lim}\ln 2^n=\underset{n \to \infty}{\lim}n\cdot(\ln 2)=\infty$ then necessarily $\underset{x\to \infty}{\lim}\ln x =\infty$. ...
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### Is my thought process in finding the tangent parallel to x & y axis correct?

I want to find the coordinates where the tangent to the curve is parallel to the x and y axis. The curve is $$2x^2 +xy - y^2 +18 = 0$$ $$Dy/dx = (-4x-y)/(x-2y)$$ Am I correct in saying that to ...
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### Prove/Disprove - Integral converges so the lim is 0

I recieved the following question and solution: 1) I didn't understand the professor's reasoning, why is the integral clearly equals to the sigma. 2) Why is it obvious that the lim dosen't exist? ...
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### I want to know if the way I derived the surface area of a sphere by integration is correct?

I am using the alias of Sillysack Buttowski and this is my first question. I searched on other links on stack exchange regarding "how to find the surface area of a sphere by integration". They seemed ...
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### How to approximate the largest eigenvalue of a monodromy matrix [closed]

Would you happen to know of a method to calculate the largest eigenvalue of a monodromy matrix? For my case the fundamental matrix cannot be calculated explicitly but it exists!
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### proving $\frac{1}{n+3}+\frac{1}{n+4}+…+\frac{1}{2n+4}>\frac{1}{2}$

how can one prove that: $\frac{1}{n+3}+\frac{1}{n+4}+...+\frac{1}{2n+4}>\frac{1}{2}$ For all natural $n$, without using induction? thank you.
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### $\left| x \right| \le 3\left[ {\sqrt x } \right]$ [closed]

Let $\left| x \right| \le 3\lfloor {\sqrt x } \rfloor$. What is the answer to this inequality?
### Role of the absolute value in $\int \frac{dx}{\sqrt{1-x^2}}$
In the derivation of the value of the indefinite integral $$\int \frac{dx}{\sqrt{1-x^2}},$$ I can substitute $x = \sin(u)$, $dx = \cos(u)du$ to get this: \...