Intro to Limits
Simple introduction to limits.
Limit Core Concept
Core concept of a limit
\[\lim_{x\to a} f(x)=L\]“when $x$ gets close to $a$, the $y$-values of the function get close to $L$”
Watch this section: what a limit means at 0:00.
Reading The Notation
The expression
\[x\to a\]means “$x$ approaches $a$.”
Graphically, we are approaching the $x$-value of $a$ from the right and left side.
If the function values ($f(x)$) from both sides approach $L$ as $x$ moves toward $a$, then the limit is $L$:
\[\lim_{x\to a} f(x)=L\]Watch this section: limit notation at 0:05.
First Example
Consider the function
\[f(x)=x+2\]What happens as our $x$-value approaches $3$ from the right and left side of our graph?
Well from the right, our values would get closer and closer to $5$ from above. We denote this as
\[\lim_{x\to 3^+}(x+2)=5 \text{ (as } x\to 3^+ \text{, + sign tells from the right, our function value }f(x) \text{ approaches }5\text{)}\]Similarly if we approach $x=3$ from the left, our values would get closer and closer to $5$ from below. We denote this as
\[\lim_{x\to 3^+}(x+2)=5 \text{ (as } x\to 3^- \text{, - sign tells from the left, our function value }f(x) \text{ approaches }5\text{)}\]Since the left-hand and right-hand behavior agree, the two-sided limit exists:
\[\lim_{x\to 3}(x+2)=5\]Watch these sections: first example at 0:28, from the left at 0:39, and from the right at 1:01.
NOTE: There are two more worked examples that one day I’ll write up in this post. Please view the video for a more comprehensive overview.
Timestamp Guide
Related Calculus Work
Use the Calculus topic page to move between limits, derivatives, and integrals.
When you are ready for the next major calculus idea, the derivatives topic page has companion guides for the power rule, product rule, chain rule, and quotient rule.