Evaluating Limits Graphically

Business Calculus · Axiom Academy

LESSON Evaluating Limits Graphically Learn to read limits directly from function graphs How to Find Limits from a Graph Looking at a graph is often the fastest way to determine limits. Here's the systematic approach: The limit is about what y-value the graph approaches , not the actual value at that point (if any). A hole or jump at x = a doesn't prevent the limit from existing! When both sides of the graph approach the same y-value, the limit exists. From the left: as x → 2⁻, the curve approaches y = 3 From the right: as x → 2⁺, the curve approaches y = 3 Note: Even though there's a hole at (2, 3), the limit still exists because both sides approach 3. Case 2: Jump Discontinuity (Limit DNE) When the left and right sides approach different values, the two-sided limit doesn't exist. From the left: _ x 1^- f(x) = 2 From the right: _ x 1^+ f(x) = 4 Since 2 ≠ 4, the one-sided limits disagree. Even when the two-sided limit doesn't exist, each one-sided limit may exist individually. Always check both directions. Case 3: Infinite Limits (Vertical Asymptotes) When the graph shoots up or down toward infinity near a point, we have an infinite limit . Reading this graph (f(x) = 1/x²) From the left: curve goes up to +∞ From the right: curve goes up to +∞ Both sides approach positive infinity. When a limit equals ∞ or -∞, mathematicians often say the limit "does not exist" (since infinity isn't a real number). But we use the ∞ notation to describe the behavior precisely.

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