Finding Limits from a Graph

Key Concepts

• One-sided limits: Limit as x approaches a value from one side (left or right).
• Limit does not exist: If the left-sided limit and right-sided limit do not match.
• Closed circle: Indicates the value of the function at that specific point.

Example 1: Limit as x Approaches 2

• Left-side limit (as x approaches 2 from the left):
  • Y value = 3
• Right-side limit (as x approaches 2 from the right):
  • Y value = -4
• Limit as x approaches 2 from either side:
  • Does not exist (left = 3, right = -4).
• Value of f(2):
  • Closed circle at x=2, Y value = 1.

Example 2: Limit as x Approaches 3

• Left-side limit (as x approaches 3 from the left):
  • Y value = 2
• Right-side limit (as x approaches 3 from the right):
  • Y value = 2
• Limit as x approaches 3 from either side:
  • Exists and is equal to 2.
• Value of f(3):
  • Closed circle at x=3, Y value = 4.

Example 3: Limit as x Approaches 4

• Left-side limit (as x approaches 4 from the left):
  • Y value ≈ -2
• Right-side limit (as x approaches 4 from the right):
  • Y value = -2
• Limit as x approaches 4 from either side:
  • Exists and is equal to -2.
• Value of f(4):
  • Closed circle at x=4, Y value = -2.

Example 4: Limit at Vertical Asymptote (x = 3)

• Left-side limit (as x approaches 3 from the left):
  • Approaches -∞.
• Right-side limit (as x approaches 3 from the right):
  • Approaches +∞.
• Limit as x approaches 3 from either side:
  • Does not exist.
• Value of f(3):
  • Not defined (no closed circle).

Example 5: Limit at Positive 4 with Asymptote

• Left-side limit (as x approaches 4 from the left):
  • Approaches +∞.
• Right-side limit (as x approaches 4 from the right):
  • Approaches +∞.
• Limit as x approaches 4 from either side:
  • Approaches +∞ (but infinity is not a number).
• Value of f(4):
  • Not defined (no closed circle).

Limits at Infinity

• Limit as x approaches positive infinity:
  • Approaches y value of the horizontal asymptote (e.g., y = 3).
• Limit as x approaches negative infinity:
  • Approaches y value of the horizontal asymptote (e.g., y = -4).
• To evaluate, look for horizontal asymptotes when x is very large.