Why do extremely high order monomials resemble exponential curves when you zoom in close to the point (1, 0)?

Question

On Desmos I found out that functions like $x^{n}$, where n is a large number, say, greater than 10, will, close to the point $(1, 0)$, resemble exponentials of the form $1/n* e^{n(x-(1-c))}$ where c is some constant that approaches 0 as n approaches infinity (I have no idea of the exact value of c for some n, I just approximated until it was really close). So presumably with a high enough n $1/n* e^{n(x-1)}$ would also be a close approximation near $(1,0)$, although maybe not considering the margin for error to be considered a "close" approximation would also decrease as n gets bigger. By $(1,0)$ I really mean some number just less than 1 and the value of the monomial at that point.