limx1

limx1 refers to the mathematical concept of the limit of a function as the variable x tends to the value 1. In formal notation, this is written as lim_{x→1} f(x) = L, where L is the value that f(x) approaches as x gets arbitrarily close to 1 within the domain of f. The limit does not necessarily equal the function value at x=1, and a limit is defined by an epsilon-delta condition: for every ε>0 there exists a δ>0 such that 0<|x−1|<δ implies |f(x)−L|<ε.

Existence of the limit can be characterized by left and right limits. The two-sided limit lim_{x→1} f(x) exists and equals L if both the left-hand limit lim_{x→1^-} f(x) and the right-hand limit lim_{x→1^+} f(x) exist and are equal to L. If the point is a boundary of the domain, only one-sided limits are defined.

Examples: (1) lim_{x→1} (x^2−1)/(x−1) = lim_{x→1} (x+1) = 2, illustrating cancellation. (2) lim_{x→1} 1/(x−1) does not exist as a finite limit; the right-hand limit is +∞ and the left-hand limit is −∞. (3) lim_{x→1} f(x) = f(1) when f is continuous at 1.

Limit operations preserve basic algebraic rules: if lim f(x)=L and lim g(x)=M, then lim [f(x)±g(x)]=L±M, lim [f(x)g(x)]=LM, and lim [f(x)/g(x)]=L/M provided M≠0. Limits underpin derivatives, integrals, and continuity analyses.