by Marilyn Traas & Albert Major
A limit describes the value that a function approaches as the input approaches a certain point. More formally, "a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value.”—It is called a limit because it represents the boundary or ultimate value that a function approaches as the input gets infinitely close to a specific point.
Simple examples of limits: These graphs show the behavior of three different functions around $x = 2$. Each of the three functions are undefined at $x = 2$ . These are just basic examples, but limits can also apply to more complex functions with different behaviors, like oscillations or approaching infinity.
We usually calculate average velocity using:
But what if the time interval is almost zero?–how fast are we going at a single point in time?
Direct calculation gives division by zero → We use limits to solve this.
This explains instantaneous velocity using secant and tangent lines. The secant line is the average velocity over an interval, but as the time interval shrinks, it approaches the tangent line, representing instantaneous velocity. If the interval becomes zero, we get $\frac{0}{0}$ $(undefined)$, which is why limits are needed. (This process also forms the foundation of derivatives, which measure how functions change at an exact moment.)
A secant is a line that intersects a curve at a minimum of two distinct points.
A tangent line is a straight line that touches a curve at exactly one point without crossing it, showing the curve’s slope at that point.
