Left & Right-Hand Limits Demystified with Graphs in PreCalculus

Learn all about limits from a graph, including left-hand limits, right-hand limits, and general limits! This video will guide you through interpreting graphical representations of functions to understand what y-values a function approaches as x gets closer to a specific point. We'll explore three detailed examples, covering continuous functions, jump discontinuities, and infinite discontinuities.

In this tutorial, you will discover:

Understanding Limit Notation:
lim (x→c⁻) f(x): The limit as x approaches c from the left side.
lim (x→c⁺) f(x): The limit as x approaches c from the right side.
lim (x→c) f(x): The general limit as x approaches c (from both sides).
Key Concept of Limits: How to determine the y-value the function is approaching, even if the function isn't defined at that exact point.
When a Limit Exists: The general limit exists if and only if the left-hand limit and the right-hand limit are equal.
Types of Discontinuities:
Jump Discontinuity: Where the left-hand and right-hand limits approach different y-values.
Infinite Discontinuity: Where the function approaches positive or negative infinity (often at a vertical asymptote).
Removable Discontinuity: A single missing point in an otherwise continuous graph.
Function Notation vs. Limit Notation: Understanding the difference between f(c) (the actual y-value at x=c) and lim (x→c) f(x) (the y-value the function is approaching).
Step-by-Step Examples:
Example 1: Continuous Function (f(x) = x² - 3) - Analyzing limits at a point where the function is continuous.
Example 2: Jump Discontinuity (f(x) = |x - 3| / (x - 3)) - Exploring left-hand, right-hand, and general limits at a jump.
Example 3: Infinite Discontinuity (f(x) = 1 / (x - 3)) - Understanding limits approaching positive and negative infinity.
This video is perfect for calculus students learning about the fundamental concept of limits and their graphical interpretation.

Timestamps:

0:00 - Introduction to Limits from a Graph
0:30 - Example 1: f(x) = x² - 3 (Continuous Function)
0:45 - Left-Hand Limit Notation (x→1⁻)
1:15 - Right-Hand Limit Notation (x→1⁺)
1:40 - General Limit (x→1) and When it Exists
2:30 - Example 2: f(x) = |x - 3| / (x - 3) (Jump Discontinuity)
3:00 - Left-Hand Limit (x→3⁻)
3:20 - Right-Hand Limit (x→3⁺)
3:40 - General Limit (x→3) and Why it Does Not Exist
4:00 - Jump Discontinuity Explained
4:20 - Bonus: Limit as x→0
4:50 - Bonus: f(3) (Function Value vs. Limit)
5:20 - Bonus: f(4)
5:40 - Example 3: f(x) = 1 / (x - 3) (Infinite Discontinuity)
6:00 - Left-Hand Limit (x→3⁻)
6:20 - Right-Hand Limit (x→3⁺)
6:40 - General Limit (x→3) and Why it Does Not Exist
7:00 - Infinite Discontinuity Explained
7:20 - Bonus: Limit as x→2
7:40 - Conclusion & More Practice
Don't forget to like, comment, and subscribe for more calculus tutorials!

#Calculus #Limits #LeftHandLimit #RightHandLimit #GeneralLimit #Discontinuity #MathTutorial #GraphingFunctions

➡️JOIN the channel as a CHANNEL MEMBER at the "ADDITIONAL VIDEOS" level to get access to my math video courses(Algebra 1, Algebra 2/College Algebra, Geometry, and PreCalculus), midterm & final exam reviews, ACT and SAT prep videos and more! (Over 390+ videos)
   / @mariosmathtutoring