If X is a Asplund space and $f:X\rightarrow \bar{\mathbb{R}}$, $C=\{x:f(x)\leq 0\}, \bar{x}\in C?

If X is a Asplund space and f:X→R, C={x:f(x)≤0},x¯∈C,f(x¯)=0C={x:f(x)≤0},x¯∈C,f(x¯)=0, is limiting subdifferential ff at x¯x¯ subset of limiting normal cone to CC at x¯x¯? i.e. ∂Lf(x¯)⊆NL(x¯,C)?∂Lf(x¯)⊆NL(x¯,C)? with which conditions it holds?