Consider the CRS production function
$$
F(K,L) = L f\left(\frac{K}{L}\right) \tag{1}
$$
Let us write $F_1 \equiv \frac{\partial F}{\partial K}$ and $F_2 = \frac{\partial F}{\partial L}$. Assume that:
$$
\begin{align*}
&\lim_{K \to 0} F_1(K,L) = \infty,\tag{a}\\
&\lim_{K \to \infty} F_1(K,L) = 0, \tag{b}\\
&\lim_{L \to 0} F_2(K,L) = \infty, \tag{c}\\
&\lim_{L \to \infty} F_2(K,L) = 0 \tag{d}
\end{align*}
$$
Notice that by differentiating $(1)$ with respect to $K$ and $L$ and defining $k = \frac{K}{L}$ we obtain:
$$
\begin{align*}
&F_1(K,L) = L f'\left(\frac{K}{L}\right) \frac{1}{L} = f'(k), \tag{A}\\
&F_2(K,L) = f\left(\frac{K}{L}\right) + L f'\left(\frac{K}{L}\right) \left(-\frac{K}{L^2}\right) = f(k) - k f'(k) \tag{B}
\end{align*}
$$
Now, consider an arbitrary sequence $(k_n)_{n \in \mathbb{N}}$. Define sequences $(K_n)_{n \in \mathbb{N}}$ and $(L_n)_{n \in \mathbb{N}}$ such that $K_n = k_n$ and $L_n = 1$. Notice that by definition $\frac{K_n}{L_n} = k_n$. Then if $k_n \to 0$ we have by $(A)$ and $(a)$:
$$
\lim_{n \to \infty} f'(k_n) = \lim_{n \to \infty} F_1(K_n, 1) = \lim_{K \to 0} F_1(K,1) = \infty.
$$
As $k_n \to 0$ was arbitrary, we have
$$
\lim_{k \to 0} f'(k) = \infty.
$$
Similarly if $k_n \to \infty$, we have by $(A)$ and $(b)$:
$$
\lim_{n \to \infty} f'(k_n) = \lim_{n \to \infty} F_1(K_n, 1) = \lim_{K \to \infty} F_1(K,1) = 0.
$$
This gives that
$$
\lim_{k \to \infty} f'(k) = 0.
$$
So far, we haven't used $(B)$ nor $(c)$ and $(d)$. These imply another set of limiting conditions on $f'(.)$. Indeed, let us define sequences $(K_n)_{n \in \mathbb{N}}$ and $(L_n)_{n \in \mathbb{N}}$ such that $K_n = 1$ and $L_n = \dfrac{1}{k_n}$. Again we have that $\frac{K_n}{L_n} = k_n$. Then for $k_n \to 0$ we have using $(B)$ and $(d)$:
$$
\lim_{n \to \infty} (f(k_n) - k_n f'(k_n)) = \lim_{n \to \infty} F_2(1,L_n) = \lim_{L_n \to \infty} F_2(1,L_n) = 0
$$
This gives that:
$$
\lim_{k \to 0} f(k_n) = \lim_{k \to 0} k f'(k).
$$
Observe that if $f$ is continuous at $0$ then this implies that:
$$
f(0) = \lim_{k \to 0} k f'(k).
$$
Next taking a sequence $k_n \to \infty$, we have using $(B)$ and $(c)$ that:
$$
\lim_{n \to \infty} (f(k_n) - k_n f'(k_n)) = \lim_{n \to \infty} F_2(1, L_n) = \lim_{L \to 0} F_2(1,L) = \infty
$$
As such,
$$
\lim_{k \to \infty} (f(k) - k f'(k)) = \infty)
$$
