Find the classes of given languages?

Question

Consider the following statements

1. $L_1 = \{wxw^R \mid w∈(a,b)^*, x∈c\}$
2. $L_2 = \{wy \mid w,y∈(a,b)^*\}$
3. $L_3 = \{zwz \mid w∈(a,b)^*,z∈\{a\}\}$
4. $L_4 = \{wxw \mid w∈(a,b)^*,x∈\{c\}^*\}$

Find the classes of languages?

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My attempt:

Obviously $L_1$ is deterministic context free language.

$L_2$ is regular language as regular expression is $(a,b)^*(a,b)^*$.

$L_3$ is also regular language as regular expression is $a(a,b)^*a$.

I'm stuck at $L_4$ it seems context sensitive language but not context free language

Can you explain in formal way, please?

Answer 1

The language $L_4$ is not context-free. Indeed, if it was, then $L_4 \cap \{a,b\}^* = \{ ww \mid w \in \{a,b\}^* \}$ would also be context-free. But it is not: see for instance this answer by Brian M. Scott.