Mathematics
Grade 12
15 min
Find one-sided limits using graphs
Find one-sided limits using graphs
What you'll learn
- Identify the missing number in an addition sentence with a sum up to 5, using pictures or objects, in at least 4 out of 5 attempts.
- Solve addition problems with sums up to 5 by drawing pictures to represent the numbers being added with 80% accuracy.
- Complete an addition sentence (e.g., 2 + __ = 4) with a sum of 5 or less, using manipulatives, with 75% accuracy.
- Draw pictures to represent and solve simple addition problems with sums up to 5.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define a one-sided limit from the left and from the right.
Interpret the mathematical notation for left-hand (x→c⁻) and right-hand (x→c⁺) limits.
Visually determine the value of a left-hand limit by tracing a function's graph.
Visually determine the value of a right-hand limit by tracing a function's graph.
Identify from a graph when a one-sided limit does not exist due to unbounded behavior (approaching ±∞).
Distinguish between the value of a one-sided limit and the value of the function at a point, f(c).
Compare left-hand and right-hand limits to draw conclusions about the existence of the overall two-sided limit.
Imagine you're hiking on a path represented by a graph. What altitude are you approaching as you get infinitely close to a c...
2
Key Concepts & Vocabulary
TermDefinitionExample
One-Sided LimitThe y-value that a function approaches as the x-value gets closer to a specific point from only one side (either from the left or from the right).The speed a car approaches as it nears a stop sign, considered only from the moments before it stops.
Left-Hand LimitThe y-value a function f(x) approaches as x gets closer to a number 'c' from values less than 'c'.Notation: lim_{x→c⁻} f(x). To find lim_{x→2⁻} f(x), we look at x-values like 1.9, 1.99, 1.999, etc.
Right-Hand LimitThe y-value a function f(x) approaches as x gets closer to a number 'c' from values greater than 'c'.Notation: lim_{x→c⁺} f(x). To find lim_{x→2⁺} f(x), we look at x-values like 2.1, 2.01, 2.001, etc.
Jump DiscontinuityA point on a graph where t...
3
Core Formulas
Left-Hand Limit Notation
lim_{x→c⁻} f(x) = L
This reads 'the limit of f(x) as x approaches c from the left is L'. The superscript minus sign (-) indicates approach from the left side (values smaller than c).
Right-Hand Limit Notation
lim_{x→c⁺} f(x) = R
This reads 'the limit of f(x) as x approaches c from the right is R'. The superscript plus sign (+) indicates approach from the right side (values larger than c).
Existence of a Two-Sided Limit
lim_{x→c} f(x) = L ⟺ lim_{x→c⁻} f(x) = L and lim_{x→c⁺} f(x) = L
The general (two-sided) limit exists and equals L if and only if the left-hand limit and the right-hand limit both exist and are equal to L.
5 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
A function f(x) is graphed. It has jump discontinuities at all integer values of x. For any integer n, lim_{x→n⁻} f(x) = n and lim_{x→n⁺} f(x) = n+1. What is the value of lim_{x→2⁻} f(x) + lim_{x→0⁺} f(x)?
A.3
B.4
C.2
D.1
Challenging
The graph of y=f(x) is provided. It has a jump at x=1, where lim_{x→1⁻} f(x) = 4 and lim_{x→1⁺} f(x) = -2. What is the value of lim_{x→0⁺} f(x+1)?
A.4
B.-2
C.Cannot be determined
D.3
Challenging
A graph of f(x) has the following properties at x=2: the curve approaches a y-value of 5 from the left; the curve approaches a y-value of -1 from the right; and there is a solid dot at (2, 5). Which statement is true?
A.The function is continuous at x=2.
B.lim_{x→2} f(x) exists.
C.The function is continuous from the left at x=2.
D.The function is continuous from the right at x=2.
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing FreeMore from Introduction to limits
Mathematics for other grades
Frequently asked questions
What grade level is "Find one-sided limits using graphs"?
Find one-sided limits using graphs is a Grade 12 Mathematics lesson on ExcelOS.
What will I learn in Find one-sided limits using graphs?
You'll be able to: Identify the missing number in an addition sentence with a sum up to 5, using pictures or objects, in at least 4 out of 5 attempts; Solve addition problems with sums up to 5 by drawing pictures to represent the numbers being….
Is "Find one-sided limits using graphs" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Find one-sided limits using graphs?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.