Infinite limits

Definition

An infinite limit exists when the value of a function increases or decreases without bound as the input approaches a certain point. This can be represented mathematically by $\lim_{{x \to a}} f(x) = \infty$ or $\lim_{{x \to a}} f(x) = -\infty$.

5 Must Know Facts For Your Next Test

1. Infinite limits occur when the function grows arbitrarily large (positive infinity) or small (negative infinity) as it approaches a certain input value.
2. The notation $\lim_{{x \to a^+}} f(x) = \infty$ describes the behavior of $f(x)$ as $x$ approaches $a$ from the right.
3. The notation $\lim_{{x \to a^-}} f(x) = -\infty$ describes the behavior of $f(x)$ as $x$ approaches $a$ from the left.
4. Vertical asymptotes are often associated with infinite limits because they represent points where the function heads towards infinity or negative infinity.
5. Infinite limits do not exist in finite terms but describe unbounded growth behavior near specific points.

Review Questions

• What does it mean for a function to have an infinite limit at a point?
• How would you mathematically express that a function $f(x)$ approaches positive infinity as x approaches some value 'a' from the right?
• Why are vertical asymptotes relevant when discussing infinite limits?

Related terms

Vertical Asymptote:

A line x=a where a function increases or decreases without bound as it approaches this line.

Finite Limits: When the value of a function approaches a specific finite number as its input nears a certain point.

$\lim_{x \to c} f(x)$: The notation used to describe the limit of function f(x) as x approaches c.