View source for Exercícios 6

'''1.''' Estude a continuidade das seguintes funções, nos respectivos domínios:

'''1.1''' $f(x)\ =  \ \left\{ \begin{array}{ll}
                                    x^2,& x\geq 0 \\
                                    & \\
                                   1, & x<0\end{array}\right.$ 

'''1.2'''  $g(x)=\left\{\begin{array}{ll}
                                    x^2,& x> 0 \\
                                    & \\
                                   1, & x<0\end{array}\right.$ 

'''1.3'''  $h(x)=\left\{\begin{array}{ll}
                                    |x|,& x\neq 0 \\
                                    & \\
                                   -1, & x=0\end{array}\right.$

'''1.4'''  $i(x)=\left\{\begin{array}{ll}
                                    x^3+x,& x\geq -1 \\
                                    & \\
                                   1-x, & x<-1\end{array}\right.$ 

'''1.5'''  $j(x)=\left\{\begin{array}{ll}
                                    2x+1,& x>3 \\
                                    & \\
                                   -x^2+3x, & x\leq 3\end{array}\right.$ 

'''1.6'''  $l(x)=\left\{\begin{array}{ll}
                                    |x+1|,& x\in [-2,0] \\
                                    & \\
                                   \frac{1}{x}, & x\in ]0,2[\\
                                   & \\
                                   x^2-4x, & x\in [2,4]\\
                                   \end{array}\right.$ 

'''2.''' Calcule, caso existam, os seguintes limites:

'''2.1''' $\displaystyle \lim_{x \to + \infty}\frac{5x^2}{x^2 +10}$

'''2.2''' $\displaystyle \lim_{x \to +\infty} \left( 1+ \frac{2}{x} \right)^x $

'''2.3''' $\displaystyle \lim_{x \to0}\frac{\cos x-1}{3x^2}$

'''2.4''' $\displaystyle \lim_{x \to +\infty}\frac{\ln(5+x)}{4+x}$

'''2.5''' $\displaystyle \lim_{x \to 2}\left(2x-\sqrt{4x^2-x}\right) $


'''2.6''' $\displaystyle \lim_{x \to 3^-}\frac{1}{3-x}$

'''2.7''' $\displaystyle \lim_{h \to 0}\frac{(a-h)^4-a^4}{h}$

'''2.8''' $\displaystyle \lim_{x \to 1}\frac{x-1}{\sqrt{(x-1)^2}}$

'''2.9''' $\displaystyle \lim_{x \to +\infty}x\left(e^{\frac{1}{x}}-1\right)$

'''2.10''' $\displaystyle \lim_{t \to 0}\frac{1-\cos(t)}{t}$

'''2.11''' $\displaystyle \lim_{x \to 0}\frac{e^x-1}{e^{2x}-1}$

'''2.12''' $\displaystyle \lim_{x \to 0}\frac{\sin\frac{x}{2}}{x}$

'''2.13''' $\displaystyle \lim_{x \to 0}[\cos(2x)]^{\frac{1}{x^2}}$

'''2.14''' $\displaystyle \lim_{x \to 0^+}\left(\frac{1}{x}\right)^x $

'''2.15''' $\displaystyle \lim_{x \to 0^+}x^{\frac{1}{\ln x}}$


'''3.''' Estude a continuidade da função
$$f(x)\ =  \ \left\{ \begin{array}{ll}
                                    \frac{1}{1+\ln x},& x\in]0,1] \\
                                    & \\
                                   0, & x=0\end{array}\right.$$
no seu domínio.

'''4.''' Determine, caso existam, as assíntotas dos gráficos de cada uma das seguintes funções:

'''4.1''' $f(x)=\displaystyle\frac{x^2-7x+10}{x-6}$;

'''4.2''' $g(x)=\ln(x^2-2x+2)$;

'''4.3''' $h(x)=\sqrt{4x^2-2x+3}$;

'''4.4''' $i(x)=e^{-\frac{1}{x}}$;

'''4.5''' $i(x)=\displaystyle\frac{\ln x}{x}$.




[[Matemática Elementar#Limites e continuidade|Voltar]]