Let $t \in \mathbb{R}_{+}^{*}$ and $x \in\ ]0,1[$. Give the limit, as $\theta$ tends to zero, of $\frac{f(t+\theta, x) - f(t, x)}{\theta}$.

Let $t \in \mathbb{R}_{+}^{*}$ and $x \in\ ]0,1[$. Show that $\lim_{h \rightarrow 0} \frac{f(t, x+h) - 2f(t, x) + f(t, x-h)}{h^{2}} = \frac{\partial^{2} f}{\partial x^{2}}(x, t)$.

Let $f : ( 0,1 ) \rightarrow \mathbb { R }$ be the function defined as $f ( x ) = \sqrt { n }$ if $x \in \left[ \frac { 1 } { n + 1 } , \frac { 1 } { n } \right)$ where $n \in \mathbb { N }$. Let $g : ( 0,1 ) \rightarrow \mathbb { R }$ be a function such that $\int _ { x ^ { 2 } } ^ { x } \sqrt { \frac { 1 - t } { t } } d t < g ( x ) < 2 \sqrt { x }$ for all $x \in ( 0,1 )$. Then $\lim _ { x \rightarrow 0 } f ( x ) g ( x )$
(A) does NOT exist
(B) is equal to 1
(C) is equal to 2
(D) is equal to 3