Finding infinite limits on a graph involves analyzing the behavior of a function as the input (x-value) approaches positive or negative infinity. You're essentially looking for horizontal asymptotes and unbounded behavior.
Understanding Infinite Limits
An infinite limit describes the behavior of a function, f(x), as x approaches positive infinity (x → ∞) or negative infinity (x → -∞). Instead of approaching a specific number, the function's value either increases without bound (approaches positive infinity) or decreases without bound (approaches negative infinity).
Steps to Find Infinite Limits on a Graph:
- Identify Potential Horizontal Asymptotes: A horizontal asymptote is a horizontal line that the graph of the function approaches as x tends towards positive or negative infinity. Visually inspect the graph far to the left and far to the right. Look for places where the graph seems to be leveling off towards a particular y-value.
- Examine the Function's Behavior as x Approaches Infinity:
- As x approaches positive infinity (x → ∞): Follow the graph to the right. Does the function approach a specific y-value (a horizontal asymptote)? If so, the limit as x approaches infinity is that y-value. Does the function increase without bound (go towards positive infinity)? Does it decrease without bound (go towards negative infinity)?
- As x approaches negative infinity (x → -∞): Follow the graph to the left. Does the function approach a specific y-value (a horizontal asymptote)? If so, the limit as x approaches negative infinity is that y-value. Does the function increase without bound (go towards positive infinity)? Does it decrease without bound (go towards negative infinity)?
- Consider Functions Approaching Zero: A key principle to keep in mind is that functions of the form 1/xn, where n is a positive number, approach 0 as x approaches positive or negative infinity. This is because as the denominator (xn) gets infinitely large, the overall fraction gets infinitely small. Look for functions that might behave similarly.
- Look for Oscillating Behavior: Sometimes, functions oscillate (wave up and down) as x approaches infinity. In these cases, the limit may not exist. However, you should still describe the oscillating behavior.
Examples:
-
Example 1: Horizontal Asymptote If the graph approaches the line y = 2 as x approaches both positive and negative infinity, then:
- lim (x→∞) f(x) = 2
- lim (x→-∞) f(x) = 2
-
Example 2: Unbounded Behavior If the graph increases without bound as x approaches positive infinity, then:
- lim (x→∞) f(x) = ∞
-
Example 3: Unbounded Behavior If the graph decreases without bound as x approaches negative infinity, then:
- lim (x→-∞) f(x) = -∞
-
Example 4: Rational Function: Consider a rational function where the degree of the numerator is less than the degree of the denominator. This will often have a horizontal asymptote at y=0. For example, f(x) = (x+1)/(x2+2) will approach 0 as x approaches positive or negative infinity.
Summary:
Finding infinite limits on a graph requires visually inspecting the function's behavior as x approaches positive and negative infinity. Look for horizontal asymptotes, unbounded growth or decay, and oscillating patterns to determine the function's limit as x tends towards infinity. Understanding how functions like 1/xn behave is crucial.
