Title: Theory / Important / Math / Analysis / Limits
On every note of this deck, unless a note says otherwise:
Page: https://en.wikipedia.org/wiki/List_of_limits

Question: \(\lim_{x\to 0}a^x=\)
Answer: 1 if \(a>0\),
\(0\) if \(a=0\)
do not exists for \(a<0\)

Question: \(\lim_{x\to \infty}\sqrt[x]{x}=\)
Answer: 1

Question: \(\lim_{x\to \infty}\left(\frac x{x+k}\right)^x=\)
Answer: \(e^{-k}\)

Question: \(\lim_{x\to0}(1+x)^{\frac1x}=\)
Answer: \(e\)

Question: \(\lim_{x\to0}(1+kx)^{\frac mx}=\)
Answer: \(e^{mk}\)
Variables: \(m,k\in\mathbb R\)

Question: \(\lim_{x\to\infty}\left(1-\frac 1x\right)^x\)
Answer: \(\frac1e\)

Question: \(\lim_{x\to0}\left(\frac{a^x-1}x\right)=\)
Answer: \(\ln a\)
Construction: \[\frac{a^x-1}x= \frac{\sum_{i\in\mathbb N}\frac{(\log ax)^i}{i!}-1}x= \frac{\sum_{i\in\mathbb N^*}\frac{(\log ax)^i}{i!}}x= \sum_{i\in\mathbb N}\frac{(\log a)^{i+1}x^i}{i!}\]

Question: \(\lim_{x\to0}\left(\frac{e^{ax}-1}x\right)=\)
Answer: \(a\)
Construction: \(\frac{e^{ax}-1}x = \frac{\sum_{i=0}^\infty\frac{(ax)^i}{i!}-1}x = \frac{\sum_{i=1}^\infty\frac{a^{i}x^{i}}{i!}}x = \frac{\sum_{i=0}^\infty\frac{a^{i+1}x^{i+1}}{(i+1)!}}x = \sum_{i=0}^\infty\frac{a^{i+1}x^i}{(i+1)!}\)

Question: \(\lim_{x\to1}\frac{\ln x}{x-1}=\)
Answer: 1

Question: \(\lim_{x\to0}x\ln x=\)
Answer: 0

Question: \(\lim_{x\to0}x\ln {\frac 1x}=\)
Answer: 0

Question: \(\lim_{x\to0}\frac{\sin ax}{bx}=\)
Answer: \(\frac ab\)

Question: \(\lim_{x\to+\infty}x\sin \frac1x=\)
Answer: 1
Construction: \[\begin{eqnarray} \lim_{x\to\infty}x\sin\frac 1x &=& \lim_{x\to\infty}x\left(\sum_{n=0}^\infty(-1)^n\frac{\left(\frac1x\right)^{2n+1}}{(2n+1)!}\right)\\ &=& \lim_{x\to\infty}x\left(\sum_{n=0}^\infty(-1)^n\frac{1}{(2n+1)!x^{2n+1}}\right)\\ & =& \lim_{x\to\infty}\sum_{n=0}^\infty(-1)^n\frac{1}{(2n+1)!x^{2n}} \\& =& \lim_{x\to\infty}\left(1+\sum_{n=1}^\infty(-1)^n\frac{1}{(2n+1)!x^{2n}}\right)\\&=&1\end{eqnarray}\]

Question: \(\lim_{x\to0}\frac{1-\cos x}x=\)
Answer: 0
Construction: \(\frac{1-\cos x}x=
\frac{1-\sum_{i=0}^{\infty}(-1)^i\frac{x^{2i}}{(2i)!}}x=
\frac{-\sum_{i=1}^{\infty}(-1)^i\frac{x^{2i}}{(2i)!}}x=
-\sum_{i=1}^{\infty}(-1)^i\frac{x^{2i-1}}{(2i)!}\)

Question: \(\lim_{x\to0}\frac{1-\cos x}{x^2}=\)
Answer: \(\frac 12\)
Construction: \(\frac{1-\cos x}{x^2} = \frac{1-\sum_{i=0}^\infty(-1)^{i} \frac{x^{2i}}{(2i)!}}{x^2} = \frac{ -\sum_{i=1}^\infty(-1)^{i} \frac{x^{2i}}{(2i)!} }{x^2} = -\sum_{i=1}^\infty(-1)^{i} \frac{x^{2(i-1)}}{(2i)!} = \sum_{i=0}^\infty(-1)^{i}\frac{x^{2i}}{(2(i+1))!} = \frac12 + \sum_{i=1}^\infty(-1)^{i+1}\frac{x^{2i}}{(2(i+1))!}\)

Question: \(\lim_{n\to \infty }\ \underbrace{\sin \sin \cdots \sin(x_0)}_n=\)
Answer: 0

Question: \(\lim_{n\to \infty }\ \underbrace{\cos \cos \cdots \cos(x_0)}_n=\)
Answer: Dottie number, the unique root of \(\cos x = x\)

Name: Dottie number
Definition: The unique solution of \(x=\cos x\)
Page: https://en.wikipedia.org/wiki/List_of_limits

Name: The Euler Mascheroni constant
Definition: \(\lim_{n\to\infty}\left(\sum_{k=1}^n\frac 1k-\log n\right)\)
Page: https://en.wikipedia.org/wiki/List_of_limits
Context: Mathematical analysis
Notation: \(\gamma\)

Question: \(\lim_{n\to\infty}(n!)^{\frac 1n}\)
Answer: \(+\infty\)

Question: \(\lim_{n\to \infty }\, 2^{n} \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+ \dots +\sqrt{2}}}}}_n\)
Answer: \(\pi\)

Question: Technics to find a limit of a function \(\lim_{x\to a}f(x)\)
Answer: Finding \(f(a)\) if it is continuous
Page: https://math.stackexchange.com/questions/1380294/list-of-techniques-to-evaluate-limits
Answer2: Simplify the expression
Answer3: Replace \(x\) by \(g(t)\) with \(\lim_{t\to b}g(t)=a\), and search \(\lim_{x\to b}f\circ g\)
In particular \(g(t)=\frac 1t\)
Answer4: Computing Taylor series at \(a\)
Answer5: L'Hôpital's rule
Answer6: Inverse function
Answer7: Showing inequalities
Answer8: Squeeze theorem

Question: Technics to find a limit of a function \(\lim_{x\to a}f(x)^{g(x)}\) when \(\lim_{x\to a}f(x)=1\) and \(\lim_{x\to a}g(x)=\infty\)
Answer: It's \(\lim_{x\to a}g(x)[f(x)-1]\)