Question

What exactly is a limit in calculus?

Answer 1

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.