Introduction to Limits
Observing the value a function approaches from the left and right sides of a point.
Introduction
What value does approach as gets closer and closer to some number? This question is the heart of calculus. A limit describes the behavior of a function near a point, even if the function is undefined at that exact point.
Prerequisite Connection
You understand function behavior, asymptotes, and graphing.
Today's Increment
We learn limit notation and evaluate limits from graphs and tables.
Why This Matters
Limits are the foundation of derivatives, integrals, and all of calculus.
Key Concepts
Limit Notation
As x approaches c, f(x) approaches L.
Left-Hand Limit
Approaching c from values less than c (from the left).
Right-Hand Limit
Approaching c from values greater than c (from the right).
Limit Exists If...
The limit exists only if left and right limits are equal:
Worked Examples
Example 1: Direct Substitution (Basic)
Find
For continuous functions, substitute directly:
Limit = 7
Example 2: Limit at a Hole (Intermediate)
Find
Direct substitution gives 0/0 (indeterminate). Factor:
Now substitute:
Limit = 4
Example 3: One-Sided Limits (Advanced)
For , find
Left:
Right:
Limit DNE (left ≠ right)
Common Pitfalls
Confusing limit with function value
The limit can exist even if f(c) is undefined or different!
Assuming 0/0 means the limit is 0
0/0 is indeterminate—you must simplify first.
Ignoring one-sided limits
Always check both sides when there's a discontinuity.
Real-World Application
Instantaneous Velocity
Your speedometer shows instantaneous speed—the limit of average velocity as the time interval approaches zero. This is exactly how derivatives work!
Practice Quiz
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