Unit 15: An Introduction to Limits
The intuitive bridge into calculus: what a function approaches, one-sided limits, continuity, and end behavior far out toward infinity.
6 topics · 22 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
The Intuitive Idea of a Limit
- How it worksHow does watching a function's output as the input creeps closer and closer to a value reveal its limit?
- Worked exampleFind the limit of f(x) = (x2-4)/(x-2) as x approaches 2 by checking values close to 2.
- Common mistakeWhy can a limit exist at a point even if the function itself is undefined right there?
- Ties togetherHow does a limit existing at a point set up the formal definition of continuity at that point?
Your turn1 of 2
Find the limit of f(x) = (x2-a2)/(x-a) as x approaches a, as a function of a.Estimating Limits from Graphs and Tables
- How it worksHow do you read a limit off a graph by tracing both sides toward the same input value?
- Worked exampleFind the limit of g(x) as x approaches 3 using the values g(2.9)=5.8, g(2.99)=5.98, g(3.01)=6.02, g(3.1)=6.2.
- Common mistakeWhy can't you find a limit just by plugging the target value straight into a table without checking both sides?
Your turn1 of 2
A function g satisfies g(7-0.1) = 6-0.1·8, g(7-0.01) = 6-0.01·8, g(7+0.01) = 6+0.01·8, g(7+0.1) = 6+0.1·8, all closing in as x approaches 7. Find the limit of g(x) as x approaches 7.One-Sided Limits
- How it worksHow do the left-hand and right-hand limits each only look at values from one side of a point?
- Worked exampleFind the left-hand and right-hand limits of f(x) = |x|/x as x approaches 0.
- Common mistakeWhy doesn't a limit exist at a point where the left-hand and right-hand limits disagree?
- Ties togetherHow does a jump between the left-hand and right-hand limits explain why a limit can fail to exist there?
Your turn1 of 2
Find the right-hand limit of f(x) = 7*|x - 3|/(x - 3) as x approaches 3 from the right.Limits That Do Not Exist
- How it worksHow do an unbounded function, a jump, and endless oscillation each cause a limit to fail to exist in its own way?
- Worked exampleFind why the limit of sin(1/x) as x approaches 0 does not exist.
- Common mistakeWhy doesn't a limit failing to exist mean the function is automatically undefined at that point?
Your turn1 of 2
Find sin(1/0.224) for 0.224 close to 0.Continuity at a Point
- How it worksHow do the three conditions of continuity check that a function's limit, value, and definition all agree at a point?
- Worked exampleFind whether f(x) = (x2-1)/(x-1) for x not equal to 1, with f(1) defined as 3, is continuous at x = 1.
- Common mistakeWhy isn't a function continuous everywhere just because you can compute a value at every point?
- Ties togetherHow does a rational function's hole from earlier in the course show up here as a point where continuity fails?
Your turn1 of 2
Find the value f(11) must be defined as for f(x) = (x2-112)/(x-11) (x not equal to 11) to be continuous at x = 11.Limits at Infinity and End Behavior
- How it worksHow does comparing the degrees of a rational function's numerator and denominator predict its limit at infinity?
- Worked exampleFind the limit of (3x2+2x)/(x2-5) as x approaches infinity.
- Common mistakeWhy doesn't a limit at infinity mean the function actually reaches infinity as a value?
- Ties togetherHow does a limit at infinity give a precise way to state the horizontal asymptote rule from earlier in the course?
Your turn1 of 2
Find the limit of (7x2+6x)/(x2-8) as x approaches infinity.