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Question Number 205727 by mathzup last updated on 28/Mar/24

$$calculate{lim}_{x\to 0}\frac{e^x-cosx}{x^2}$$

Commented by lepuissantcedricjunior last updated on 29/Mar/24

$$\underset{x\to 0}{\lim }\frac{{\mathit{e}}^{\mathit{x}}-\mathit{c}\mathit{o}\mathit{s}\mathit{x}}{{\mathit{x}}^2}=\frac{0}{0}=\mathbf{FI}$$$$\iff \underset{x\to 0}{\lim }\frac{{\mathit{e}}^{\mathit{x}}-1-(\mathit{c}\mathit{o}\mathit{s}\mathit{x}-1)}{{\mathit{x}}^2}$$$$\iff \underset{x\to 0}{\lim }(\frac{{\mathit{e}}^{\mathit{x}}-1}{{\mathit{x}}^2}-\frac{\mathit{c}\mathit{o}\mathit{s}\mathit{x}-1}{{\mathit{x}}^2})$$$$\mathit{d}\mathit{l}{\mathit{e}}^{\mathit{x}}=1+\mathit{x}+\frac{{\mathit{x}}^2}{2!}+\frac{{\mathit{x}}^3}{3!}+{\mathit{x}}^3\left(0\right)$$$$\mathit{d}\mathit{l}\mathit{c}\mathit{o}\mathit{s}\left(\mathit{x}\right)=1-\frac{{\mathit{x}}^2}{2!}+\frac{{\mathit{x}}^4}{4!}+{\mathit{x}}^4\left(0\right)$$$$\underset{x\to 0}{\lim }\frac{1+\mathit{x}+\frac{{\mathit{x}}^2}{2}+\frac{{\mathit{x}}^3}{3!}-1-1+\frac{{\mathit{x}}^2}{2}-\frac{{\mathit{x}}^4}{4}+1}{{\mathit{x}}^2}$$$$\iff \underset{x\to 0}{\lim }\frac{1+\mathit{x}+\frac{{\mathit{x}}^2}{3!}-\frac{{\mathit{x}}^3}{4!}}{\mathit{x}}=\frac{1}{0}=\mathrm{\infty }$$$$\underset{x\to 0}{\lim }\frac{{\mathit{e}}^{\mathit{x}}-\mathit{c}\mathit{o}\mathit{s}\mathit{x}}{{\mathit{x}}^2}=\mathrm{\infty }$$$$.............lepuissantDr......................$$

Commented by mr W last updated on 28/Mar/24

$$pleasepostyouranswertoaquestion$$$$as‘‘{answer}^{″},notas‘‘{comment}^{″}!thanks!$$

Commented by mr W last updated on 28/Mar/24

veuillez poster votre réponse à une question comme "réponse", pas comme "commentaire"! Merci!

Commented by York12 last updated on 29/Mar/24

$$$$Tu es Francis!?

Answered by mr W last updated on 28/Mar/24

$$=\underset{x\to 0}{\lim }\frac{e^x+\sin x}{2x}=\frac{1}{0}=\mathrm{\infty }$$

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