Questions tagged [limits]
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Questions on the evaluation and properties of limits in the sense of analysis and related fields. For limits in the sense of category theory, use (limits-colimits) instead.
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Evaluating $\lim_{x\to 0}\frac{4\sin x-\sin 4x}{x^3}$ without L'Hopital's Rule [closed]
Could someone explain this limit without L'Hopital's Rule, please! $$\lim_{x\to 0}\frac{4\sin x-\sin 4x}{x^3}$$ calculus limits limits-without-lhopital- 1
Calculate the total derivative
Calculate directly the total derivative (without using partial derivatives) of the function $f(x_1,x_2)=x_1^2-10x_2$. You may wish to use the fact that $\sqrt{x^2+y^2}\geq\frac{x+y}{2}.$ $$$$ I have ... limits multivariable-calculus derivatives- 59
Can we use $(1+x)^n = 1+nx$ where $x\to0$ and $n$ is $1/0$?
I was solving a limits questions: $$\lim_{x\toπ/4} \tan x^{\tan2x}$$ After putting $x = (π/4)+h$ and solving it I got the expression: $$(1-h/1+h)^{\cot2x} = (1-2h)^{\cot2h}$$ $$(1-2h)^{\cot2h}$$ Now ... limits limits-without-lhopital binomial-theorem- 11
How do we go from $x^n - a^n$ to $(x-a)(x^{n-1}+ ax^{n-2} +\ldots + xa^{n-2} + a^{n-1})$
This is involved in the proof of the standard limit x tends to a:$\dfrac{(x^n - a^n)}{(x - a)}$. How can we prove this statement and how do we know that the second term ends at $x^0$. Is this limit ... algebra-precalculus limits regular-expressions- 1
Limit of $a_{n+1}=a_{n}\cdot \left( 1+\frac{1}{n}\right)$
Let $a_{n}$ be a recursive sequence such as: $\begin{cases}a_{1}=1\\ a_{n+1}=a_{n}\cdot \left( 1+\dfrac{1}{n}\right) \end{cases}$ I need to show that $\lim _{n\rightarrow \infty }a_{n}=\infty$. I ... sequences-and-series limits infinite-product- 61
Proving $ \lim_{x \to 0^{+}} 4^{\frac{1}{x}} = \infty $
I want to prove that $ \lim_{x \to 0^{+}} 4^{\frac{1}{x}} = \infty $, because I can't isolate $ x $ without flipping the $ \gt $ sign. Here is my proof so far: Given $ M > 0 $, Choose $ \delta = $ ... real-analysis limits- 89
Calculate total derivative directly.
Calculate directly (not via partial differentiation) the total derivative of the function $f(x_1,x_2)=x_1^2-10x_2.$ You may wish to use the fact that $\sqrt{x^2+y^2}\geq\frac{x+y}{2}.$ $$$$ For the ... limits multivariable-calculus derivatives multivalued-functions- 59
Prove by definition $\lim _{n\rightarrow \infty }\frac{1}{a_{n}}=\frac{1}{L}$ [closed]
maybe somebody could help me to prove the following statement: Let $a_{n}$ be a sequence. Assume $\lim _{n\rightarrow \infty }a_{n}=L$ and that $L\neq 0$. Prove by definition that $\lim _{n\rightarrow ... sequences-and-series limits- 61
Spivak's Calculus, Chapter 5 "Limits", problem 41: If $c>0$, then $c^{1/n}$ approaches $1$ as $n$ > becomes very large. [duplicate]
The following is a problem from chapter 5, "Limits", of Spivak's Calculus (a) For $c>1$, show that $c^{1/n}=\sqrt[n]{c}$ approaches $1$ as $n$ becomes very large. Hint: Show that for any ... calculus limits solution-verification proof-explanation- 1,865
I not understanding a cancellation step in Spivak proof.
There is a question about this exact same proof but the answer is not satisfying me so I'm going to run it again if you don't mind. I will post the original question after mine. In his chapter on ... calculus algebra-precalculus limits inequality absolute-value- 1,292
Proving that if derivative of f(x) = a with a>0, f(x) must go to infinity
I am interested in proving that if derivative of f(x) is a real number c, c>0, as x goes to infinity, f(x) itself must go to infinity. Seems like a common-sense statement, but don't know how I can ... real-analysis limits derivatives- 13
How do I compute this limit: $\sum_{k = 1}^{\infty} \frac{\sin k}{k}$? [duplicate]
My task is to compute the limit of $$\sum_{k = 1}^{\infty} \dfrac{\sin k}{k}$$, with Fourier-theory. The only thing I know is that $$\dfrac{\sin k}{k}$$ are the coefficients of the Fourier series $$\... calculus limits convergence-divergence fourier-series- 11
If $\lim_{x\to1} \frac{f(x)-7}{x-1}=12$, find $\lim_{x\to1} f(x)$? [closed]
If $\displaystyle \lim_{x\to1} \frac{f(x)-7}{x-1}=12$, find $\displaystyle \lim_{x\to1} f(x)$? How do I solve this limits problem? calculus limits- 1
Taylor series expansion of a function which is undefined at a point
I have a function $f(x)$ which is well defined and differentiable in the whole $\mathbb{R^+}$, positive section of real line, except it is undefined at the origin $x=0$. I want to show that near the $... calculus limits functions- 2,829
In an equilateral triangle, infinite line segments connect a vertex to the opposite side. If the product of the lengths converges and >0, what is it?
In an equilateral triangle, line segments connect a vertex to $n$ uniformly distributed points on the opposite side, including points at the ends. Assuming $\lim_{n\to\infty}\text{(product of lengths ... real-analysis geometry limits triangles infinite-groups- 1,324
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