What is the purpose of a limit in calculus?

Question

Emma Aldridge · Accepted Answer

A limit allows us to examine the tendency of a function around a given point even when the function is not defined at the point.  Let us look at the function below. 
#f(x)={x^2-1}/{x-1}#
Since its denominator is zero when #x=1#, #f(1)# is undefined; however, its limit at #x=1# exists and indicates that the function value approaches #2# there.
#lim_{x to 1}{x^2-1}/{x-1}
=lim_{x to 1}{(x+1)(x-1)}/{x-1}
=lim_{x to 1}(x+1)=2# This tool is very helpful in calculus when the definition of the derivative is motivated by the fact that the slopes of secant lines with approaching intersection points approximate the slope of a tangent line.

Adam Adkins · Answer

undefined The purpose of a limit in calculus is to determine the behavior of a function as it approaches a particular value or point. It helps in analyzing the continuity, differentiability, and convergence of functions. Limits are used to define derivatives and integrals, and they play a crucial role in understanding the fundamental concepts of calculus.