Introduction to Limits

Key Questions

  • 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.

  • 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.

Questions