Dadas as funções reais de variável real
Teoria de limites: Infinito 12 A - Parte 2 Pág. 207 Ex. 21
Dadas as funções reais de variável real, assim definidas:$$\begin{array}{*{20}{c}}
{f(x) = {x^2} + 1}&{\text{e}}&{g(x) = \frac{1}{x}}
\end{array}$$
- Determine, em função de $h$, a taxa média de variação de cada uma das funções no intervalo $\left[ {1,1 + h} \right]$, com $h > 0$.
- Calcule se existir:
a) $\mathop {\lim }\limits_{h \to 0} \frac{{f(1 + h) – f(1)}}{h}$
b) $\mathop {\lim }\limits_{h \to 0} \frac{{g(1 + h) – g(1)}}{h}$
c) $\mathop {\lim }\limits_{h \to 0} \frac{{f(a + h) – f(a)}}{h}$
d) $\mathop {\lim }\limits_{h \to 0} \frac{{g(a + h) – g(a)}}{h}$, para ${a \ne 0}$
$$\begin{array}{*{20}{c}}
{f(x) = {x^2} + 1}&{\text{e}}&{g(x) = \frac{1}{x}}
\end{array}$$
- Para $f$:
$$\begin{array}{*{20}{l}}
{tm{v_{\left[ {1,1 + h} \right]}}}& = &{\frac{{f(1 + h) – f(1)}}{h}} \\
{}& = &{\frac{{{{(1 + h)}^2} + 1 – 2}}{h}} \\
{}& = &{\frac{{1 + 2h + {h^2} – 1}}{h}} \\
{}& = &{2 + h}
\end{array}$$
Para $g$:
$$\begin{array}{*{20}{l}}
{tm{v_{\left[ {1,1 + h} \right]}}}& = &{\frac{{g(1 + h) – g(1)}}{h}} \\
{}& = &{\frac{{\frac{1}{{1 + h}} – 1}}{h}} \\
{}& = &{\frac{{\frac{{1 – 1 – h}}{{1 + h}}}}{h}} \\
{}& = &{ – \frac{1}{{1 + h}}}
\end{array}$$ -
a)
Aproveitando o resultado obtido acima, vem:
$$\begin{array}{*{20}{l}}
{\mathop {\lim }\limits_{h \to 0} \frac{{f(1 + h) – f(1)}}{h}}& = &{\mathop {\lim }\limits_{h \to 0} \left( {2 + h} \right)} \\
{}& = &2
\end{array}$$
b)
Aproveitando o resultado acima, vem:
$$\begin{array}{*{20}{l}}
{\mathop {\lim }\limits_{h \to 0} \frac{{g(1 + h) – g(1)}}{h}}& = &{\mathop {\lim }\limits_{h \to 0} \left( { – \frac{1}{{1 + h}}} \right)} \\
{}& = &{ – 1}
\end{array}$$
c)
$$\begin{array}{*{20}{l}}
{\mathop {\lim }\limits_{h \to 0} \frac{{f(a + h) – f(a)}}{h}}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{{{\left( {a + h} \right)}^2} + 1 – \left( {{a^2} + 1} \right)}}{h}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{{a^2} + 2ah + {h^2} + 1 – {a^2} – 1}}{h}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{2ah + {h^2}}}{h}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \left( {2a + h} \right)} \\
{}& = &{2a}
\end{array}$$
d) para ${a \ne 0}$,
$$\begin{array}{*{20}{l}}
{\mathop {\lim }\limits_{h \to 0} \frac{{g(a + h) – g(a)}}{h}}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{\frac{1}{{a + h}} – \frac{1}{a}}}{h}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \frac{{\frac{{a – a – h}}{{a\left( {a + h} \right)}}}}{h}} \\
{}& = &{\mathop {\lim }\limits_{h \to 0} \left( { – \frac{1}{{a\left( {a + h} \right)}}} \right)} \\
{}& = &{ – \frac{1}{{{a^2}}}}
\end{array}$$
