Limits and Continuity

Limits

• If the function is continuous at the point or has a removable discontinuity there, (a hole where the function is continuous if the hole were filled), the limit exists.

  • You may only need to plug in the requested value into the variable to find the answer. Otherwise, you may need to use Squeeze Theorem, L’Hopital’s rule, or another method as detailed here.

• If the function has a vertical asymptote at that point, the function may exist or may not exist.

  • If the function on both sides of the asymptote approaches either positive or negative infinity, the limit exists and evaluates to that positive or negative infinity.

  • If the function approaches positive infinity on one side of the asymptote and negative infinity on the other, the limit does not exist.

• If the function is piecewise and noncontinuous at the point, the limit does not exist.

Continuity

1. The function must be defined at the point in question.

2. The limit of the function as x approaches the point in question must exist.

3. The limit of the function as x approaches the point in question must be equal to the value of the function at the point in question.

If all three conditions are met, then the function is continuous at that point.

Intermediate Value Theorem

Go here for a detailed explanation of the Intermediate Value Theorem.