MicroMaths | Maths by New Technologies

Challenge

Inequality - Derivation

Using the Derivation Concept, Prove that :

${\forall(a,b)\in\mathbb{R}_+^{\large\ast}^2 \qquad : \qquad \,\sqrt{a+b}(\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}})\geq2\sqrt{2}}$

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Calculus of a Limit

Calculate:

${\lim_{x\rightarrow 0}\frac{1-tan(\frac{\pi}{4}+x)tan(\frac{\pi}{4}+2x)tan(\frac{\pi}{4}-3x)}{x^{3}}}$

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Application of I.V.T

Let $f$ be a numerical function defined on $I=[0,1]$ , as:

$f$[0,1]$ \subset ]0,1[$

Prove that:

$\forall n \in \mathbb{N} : \qquad \exist x_{n} \in ]0,1[ \qquad / \qquad f$x_n$=(x_n)^n$

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I.V.T and Limits at Infinity

Let $f$ be a continuous function on $\mathbb{R}$ .
Proove that :

$\big(\qquad \lim_{x \to +\infty}f(x)=-\infty \qquad and \qquad \lim_{x \to -\infty}f(x)=+\infty \qquad \big)\Rightarrow \big( \exist \alpha \in \mathbb{R} \qquad / \qquad f(\alpha)=0\big)$

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Calculus of a Limit

Calculate this limit :

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Read more

Calculus of a Limit

Calculate:

${\lim_{x\rightarrow 0}\frac{1-tan(\frac{\pi}{4}+x)tan(\frac{\pi}{4}+2x)tan(\frac{\pi}{4}-3x)}{x^{3}}}$

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Inequality - Derivation

Using the Derivation Concept, Prove that :

${\forall(a,b)\in\mathbb{R}_+^{\large\ast}^2 \qquad : \qquad \,\sqrt{a+b}(\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}})\geq2\sqrt{2}}$

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Henri Poincaré

Poll

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