Matemática A - 12.º - aulas e exercícios em vídeo e fichas multimédia

5.1) Com exceção da 5.1a), a resolução é reservada a inscritos. Inscreva-se neste link!

Calcule cada um dos seguintes limites:

(a) $\displaystyle\mathop {\lim }\limits_{x \to 0} \frac{{{e^x} - 1}}{{2x}}$;	(b) $\displaystyle\mathop {\lim }\limits_{x \to 0} \frac{{1 - {e^{3x}}}}{{2x}}$;	(c) $\displaystyle\mathop {\lim }\limits_{x \to 0} \frac{{{e^{ - x}} - 1}}{x}$;
(d) $\displaystyle\mathop {\lim }\limits_{x \to 0} \frac{{{e^x} - {e^{5x}}}}{{2x}}$;	(e) $\displaystyle\mathop {\lim }\limits_{x \to 0} \frac{{{e^x} - 1}}{{{e^{2x}} - 1}}$;	(f) $\displaystyle\mathop {\lim }\limits_{x \to 3} \frac{{{e^x} - {e^3}}}{{x - 3}}$;
(g) $\displaystyle\mathop {\lim }\limits_{x \to  + \infty } {e^x}$;	(h) $\displaystyle\mathop {\lim }\limits_{x \to  + \infty } {\pi ^x}$;	(i) $\displaystyle\mathop {\lim }\limits_{x \to  - \infty } {e^{3x}}$;
(j) $\displaystyle\mathop {\lim }\limits_{x \to  + \infty } {\left( {\frac{2}{3}} \right)^x}$;	(l) $\displaystyle\mathop {\lim }\limits_{x \to  + \infty } \frac{{{2^x}}}{{{e^x}}}$;	(m) $\displaystyle\mathop {\lim }\limits_{x \to  - \infty } \frac{{{2^x}}}{{{e^x}}}$;
(n) $\displaystyle\mathop {\lim }\limits_{x \to  + \infty } \frac{{{e^x}}}{{2x}}$;	(o) $\displaystyle\mathop {\lim }\limits_{x \to  + \infty } \frac{{{e^{2x}}}}{{{x^3}}}$;	(p) $\displaystyle\mathop {\lim }\limits_{x \to  + \infty } \frac{x}{{{e^{2x}}}}$.

5.2) Resolução reservada a inscritos. Inscreva-se neste link!

Calcule cada um dos seguintes limites:

(a) $\displaystyle\mathop {\lim }\limits_{x \to 0} \frac{{\ln {{\left( {x + 1} \right)}^2}}}{{4x}}$;	(b) $\displaystyle\mathop {\lim }\limits_{x \to 0} \frac{{\ln \left( {2x + 1} \right)}}{{3x}}$;	(c) $\displaystyle\mathop {\lim }\limits_{x \to \frac{1}{2}} \frac{{\ln \left( {2x} \right)}}{{2x - 1}}$;
(d) $\displaystyle\mathop {\lim }\limits_{x \to  + \infty } \frac{{\ln \left( {2x} \right)}}{x}$;	(e) $\displaystyle\mathop {\lim }\limits_{x \to  - \infty } \frac{{\ln \left( { - 3x} \right)}}{x}$;	(f) $\displaystyle\mathop {\lim }\limits_{x \to  - \infty } \frac{{{x^5}}}{{\ln \left( { - x} \right)}}$;