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^21}/{x1}#
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^21}/{x1} =lim_{x to 1}{(x+1)(x1)}/{x1} =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^21}/{x1}#
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^21}/{x1} =lim_{x to 1}{(x+1)(x1)}/{x1} =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
Limits

Introduction to Limits

Determining One Sided Limits

Determining When a Limit does not Exist

Determining Limits Algebraically

Infinite Limits and Vertical Asymptotes

Limits at Infinity and Horizontal Asymptotes

Definition of Continuity at a Point

Classifying Topics of Discontinuity (removable vs. nonremovable)

Determining Limits Graphically

Formal Definition of a Limit at a Point

Continuous Functions

Intemediate Value Theorem

Limits for The Squeeze Theorem