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