Relationship between Chebyshev T and Gegenbauer C

Theorem

The following formula holds for $m,n \in \{0,1,2,\ldots\}$: $$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda}; n\geq 1,$$ where $T_n$ denotes a Chebyshev polynomial of the first kind and $C_n^{\lambda}$ denotes a Gegenbauer C polynomial.