Lista 9
Zadanie 1

$$
f(x)
=
\begin{cases}
\frac{e^{7x}-1}{x} & \text{ dla } x≠0\\
7 & \text{ dla } x =0
\end{cases}
$$
$$\lim_{x\rightarrow0^{+-}} \frac{e^{7x} - 1}{x} =\lim_{x\rightarrow0^{+-}} \frac{(e^{7x} - 1)'}{x'}=\lim_{x\rightarrow0^{+-}} \frac{7e^{7x}}{1}=7$$

$$\left(\frac{e^{7x} - 1}{x}\right)'=\frac{(e^{7x} - 1)'x-(e^{7x} - 1)\cdot x'}{x^2}$$

\begin{align*}
\lim_{h \to 0} \frac{f(h) - f(0)}{h}
&=
\lim_{h \to 0} \frac{\frac{e^{7h}-1}{h} - 7}{h}\\
&=
\lim_{h \to 0} \frac{e^{7h}-1 - 7 h }{h^2}\\
&=
\lim_{h \to 0} \frac{49e^{7h}}{2} = \frac {49}{2}
\end{align*}