Calculus

by Marilyn Traas & Albert Major

Limits in more detail

Limits Involving Infinity

• Infinity is not a number but a concept.
• If both the left-hand and right-hand limits tend toward the same type of infinity (either +∞ or ∞), then the limit exists.
• If one side goes to +∞ while the other goes to ∞, the limit does not exist.

When a Limit Exists and When It Doesn't

A limit exists when:

• The left-hand and right-hand limits are equal at a point.
• The function approaches a finite value, even if it is not defined at that point.
• In cases involving infinity, both sides must tend toward the same type of infinity (either +∞ or ∞).
• For rational functions, if the denominator does not tend to zero at the given x-value, the limit is well-defined.

A limit does not exist when:

• The left-hand and right-hand limits are different.
• The function oscillates infinitely near a point (e.g., sin(1/x) as x → 0).
• The function grows infinitely large in different directions from different sides (e.g., one side goes to +∞, the other to ∞).
• The function has a vertical asymptote at that point (approaching infinite values).

Limit Rules