Title: Maths / Chapitre 1.B. - Premières fonctions usuelles

Q: \(\lim_{x\to +\infty}\frac{\ln x}{x}=\)
A: \(\lim_{x\to +\infty}\frac{\ln x}{x}=0\)

Q: \(\lim_{x\to 1}\frac{\ln x}{x-1}=\)
A: \(\lim_{x\to 1}\frac{\ln x}{x-1}=1\)

Q: \(\lim_{h\to 0}\frac{\ln(1+h) }{h}=\)
A: \(\lim_{h\to 0}\frac{\ln(1+h) }{h}=1\)

Q: \(\lim_{x\to 0}x\ln x=\)
A: \(\lim_{x\to 0}x\ln x=0\)

Q: \(\lim_{x\to 0}\frac{\exp(x)-1}{x}=\)
A: \(\lim_{x\to 0}\frac{\exp(x)-1}{x}=1\)    (car : \(\frac{\exp(x)-\exp(0)}{x-0}\xrightarrow[x \to \infty]{}\exp'(0)=\exp(0)=1\))

Q: \(\lim_{x\to \infty}\frac{\exp(x)}{x}=\)
A: \(\lim_{x\to \infty}\frac{\exp(x)}{x}=+\infty\)

Q: \(\lim_{x\to -\infty}x\exp(x)=\)
A: \(\lim_{x\to -\infty}x\exp(x)= 0\)

Q: Soit \(\alpha\in \mathbb{R},\,\forall \, x\in \mathbb{R_+^*},\, x^\alpha =\)[...]
A: \(\exp(\alpha \ln(x))\)

Q: \(\forall \, x \in \mathbb{R} \setminus \left\{ \frac{\pi}{2}+k\pi /k \in \mathbb{Z}\right\}, \tan'(x) =\)[...][...]
A: \(1+\tan^2 x=\); \(\frac{1}{\cos^2 x}\)

Q: \(\cos(x)=0 \iff\)[...]
A: \(x=\frac{\pi}{2}[\pi]\)

Q: \(\sin(x)=0 \iff\)[...]
A: \(x=0\,[\pi]\)

Q: \(\tan(x)=0 \iff\)[...]
A: \(x=0\,[\pi]\)

Q: \(\cos x = \cos y \iff \)[...]
A: \(x=\pm \,y\,[2\pi] \)

Q: \(\tan x = \tan y \iff \)[...]
A: \(x=y\,[2\pi] \) ou \(x=y+\pi [2\pi]\)

Q: \(\sin x = \sin y \iff\)[...]
A: \(x=y\,[2\pi]\)ou\(x=\pi-y\,[2\pi]\)

Q: \(\lim_{x \to 0} \frac{\sin(x)}{x}=\)
A: \(\lim_{x \to 0} \frac{\sin(x)}{x}=1\)

Q: \(\tan(a+b) = \)
A: \(\frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}\)=

Q: \(\tan(a-b)=\)
A: \(\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}=\)

Q: \(t=\tan\left(\frac{x}{2}\right)\)
\(\tan x =\)
A: \(\frac{2t}{1-t^2}\)

Q: \(t=\tan\left(\frac{x}{2}\right)\)
\(\sin x=\)
A: \(\frac{2t}{1+t^2}\)

Q: \(t=\tan\left(\frac{x}{2}\right)\)
\(\cos x =\)
A: \(\frac{1-t^2}{1+t^2}\)

Q: \(\cos(a)\cos(b)=\)
A: \(\frac{1}{2}\left(\cos(a+b)+\cos(a-b)\right)=\)

Q: \(\sin(a)\sin(b)=\)
A: \(\frac{1}{2}\left(\cos(a-b)-\cos(a+b)\right)=\)

Q: \(\sin(a)\cos(b)=\)
A: \(\frac{1}{2}\left(\sin(a+b)+\sin(a-b)\right)=\)

Q: \(\cos(p)+\cos(q)=\)
A: \(2\cos\left(\frac{p+q}{2}\right)\cos\left(\frac{p-q}{2}\right)=\)

Q: \(\cos(p)-\cos(q)=\)
A: \(-2\sin\left(\frac{p+q}{2}\right)\sin\left(\frac{p-q}{2}\right)=\)

Q: \(\sin(p)+\sin(q)=\)
A: \(2\sin\left(\frac{p+q}{2}\right)\cos\left(\frac{p-q}{2}\right)=\)

Q: \(\sin(p)-\sin(q)=\)
A: \(2\cos\left(\frac{p+q}{2}\right)\sin\left(\frac{p-q}{2}\right)=\)