Prove the limit exists if and only if the left and right limits exist and are equal

Calculate the limit of the following function if it exists

\lim\limits_{x\to 0} \dfrac{|x|}{x}

a_1=3 , a_n=2a^2_{n-1}-1 , find the value of \lim\limits_{n\to \infty}\dfrac{a_n}{2^na_1a_2\dots a_{n-1}}

If n\in N^* such that a_1=\sqrt{2} , a_{n+1}=\sqrt{2+a_n}, show that a_n has limit and find the value of the the limit.

Find the limit

\lim\limits_{n\to \infty} \dfrac{6n+75}{2n+109}