ε, δ Conditions for One-sided Limits

There are also ε, δ conditions for one-sided limits and infinite limits. The three cases below are typical.

ε, δ CONDITION FOR lim_x→c f(x) = L

For every real number ε > 0, there is a real number δ > 0 which depends on e such that whenever c < x < c + δ, we have |f(x) - L| < ε.

Intuitively, when x is close to c but greater than c, f(x) is close to L.

ε, δ CONDITION FOR lim_x→∞ f(x) = L

For every real number ε > 0 there is a real number B > 0 which depends on ε. such that whenever x > B, we have |f(x) - L| < ε.

Intuitively, when x is large, f(x) is close to L.

ε, δ CONDITION FOR lim_x→∞ f(x) = ∞

For every real number A > 0 there is a real number B > 0 which depends on A such that whenever x > B, we have f(x) > A.

Intuitively, when x is large, f(x) is large.

Example 3

Example 4