Ensino Superior ⇒ Limites - sem usar L'Hospital

[magben]

[tex3]\lim_{x \rightarrow 0} \frac{x^2}{1-\sec x}[/tex3]

[ti123]

Coloque o gabarito, caso o tenha. Por favor

[AlexandreHDK]

[tex3]\lim_{x \to 0}\frac{x^2}{1-\sec x}=\lim_{x \to 0}\frac{x^2}{1-\frac{1}{\cos x}}=\lim_{x \to 0}\frac{x^2 \cos x}{ \cos x-1}[/tex3]
[tex3]=-\lim_{x \to 0}\frac{x^2 \cos x}{1 - \cos x} \frac{1+\cos x}{1 +\cos x}=-\lim_{x \to 0}\frac{x^2 \cos x (1+\cos x)}{1 - \cos^2 x}=-\lim_{x \to 0}\frac{x^2 \cos x (1+\cos x)}{\sen^2 x}[/tex3]
[tex3]=-\lim_{x \to 0}\frac{\cos x (1+\cos x)}{\frac{ \sen^2 x}{x^2}}=-\frac{\lim_{x \to 0}\cos x (1+\cos x)}{\lim_{x \to 0}\frac{\sen^2 x}{x^2}}[/tex3]
[tex3]=-\frac{\cos(0).(1+\cos(0))}{1}=-\frac{1.(1+1)}{1}=-2[/tex3]