What exactly is a limit in calculus?

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Mar 1, 2018

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 useful in calculus when the slope of a tangent line is approximated by the slopes of secant lines with nearing intersection points, which motivates the definition of the derivative.

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