Calculus · Grades 11 to 12
Limits
Find the value a function approaches.
The idea
A limit asks what value a function approaches as x gets close to a point. The function does not have to exist at the point itself.
If direct substitution gives 0/0, factor the top and cancel the common factor. Then substitute again.
Worked examples
- How to Find a Limit by Factoring Evaluate the limit of (x^2 - 9)/(x - 3) as x goes to 3. Four steps, the 0/0 case, the cancel mistakes to avoid and practice problems.
- How to Evaluate a Limit by Substitution Evaluate the limit of x^2 + 3x as x goes to 2. Three steps with direct substitution, when it is allowed, and the arithmetic slips to avoid.
- How to Evaluate a Limit by Rationalizing Evaluate the limit of (sqrt(x + 9) - 3)/x as x goes to 0 with the conjugate. Five steps, the 0/0 case, and the cancel mistakes to avoid.
Practice this topic
4 problems with answers.