Ensino Superior ⇒ Limite Trigonométrico Tópico resolvido

[demetrius]

Calcule se existir limite de [tex3]\frac{sen(x)-cos(x)}{1-tg(x)}[/tex3] quando x tende a [tex3]\frac{\pi}{4}[/tex3]

[Karl Weierstrass]

Calcule se existir [tex3]\lim_{x\rightarrow \frac{\pi}{4} }\,\,\, \frac{\text{sen}\,x\,-\,\cos\,x}{1-\text{tg}x} [/tex3]

[tex3]\hspace{70pt}\lim_{x\rightarrow \frac{\pi}{4} }\,\,\, \frac{\text{sen}\,x\,-\,\cos\,x}{1-\text{tg}x} =

\frac{\text{sen}\, \frac{\pi}{4} \,-\,\cos\, \frac{\pi}{4} }{1-\text{tg} \frac{\pi}{4} } =

\frac{0}{0} [/tex3]

[tex3]\hspace{70pt}\lim_{x\rightarrow \frac{\pi}{4} }\,\,\, \frac{\text{sen}\,x\,-\,\cos\,x}{1- \frac{\text{sen}\,x}{\cos\,x}} =

\lim_{x\rightarrow \frac{\pi}{4} }\,\,\, \frac{-\cancel{(\cos\,x-\text{sen}\,x)}}{ \frac{\cancel{\cos\,x\,-\,\text{sen}\,x}}{\cos\,x}} \,=\,\lim_{x\rightarrow \frac{\pi}{4} }\,\,- \cos x=-\cos\, \frac{\pi}{4} =- \frac{\sqrt{2}}{2} .[/tex3]

Também pode ser resolvido pelo Teorema de L'Hôspital.






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