12.1 (cont.) - Introduction to Limits

Properties of Limits:
Let b and c be real numbers and let n be a positive integer.












Operations with Limits:
Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:
1. Scalar multiple:





2. Sum or difference:




3. Product:




4. Quotient:





5. Power:




Direct Substitution:
You have seen that sometimes the limit of f(x) as x approaches c is simply f(c). In such cases, it is said that the limit can be evaluated by direct substitution.Finding the limit using direct substitution is fairly easy. Just plug the c-value into the function and then solve using simple algebra skills.

Continuity:
A graph is discontinuous if there are any:
- holes
- asymptotes
- breaks
Here, f is continuous at a.

T or F?
If...




then f is continuous at a.

This is FALSE.
This proves that there is in fact a limit, but it's not for certain that the graph is continuous.