Unit 5 Summary

Calculus Readiness · Axiom Academy

SUMMARY Unit 5 Summary: Limits Your first taste of calculus — the idea that a function can approach a value even if it never reaches it. A limit describes behavior near a point, not the value at it — f(a) need not even exist. You can find a limit three ways: from a graph , from a table of nearby values, and algebraically . When the limit exists, all three agree. A two-sided limit exists only when the left and right one-sided limits agree . If f is continuous at a , then — direct substitution is always the first move. Hitting doesn't mean "no answer" — factor, cancel, or rationalize, then substitute again. Limits are the bridge to all of calculus : every derivative and every integral is secretly a limit. As x gets arbitrarily close to a from either side , f(x) gets arbitrarily close to L . The limit describes where the function is headed , not where it lands. Key idea: the value f(a) is irrelevant — and may not even exist. Watch out for: "approaches" "equals." Behavior near a , not value at a . Core Concept Limits from Graphs & Tables Read a limit off a graph by tracing the curve toward x = a from both sides, or off a table by plugging in values that close in on a (e.g. ). When to use: early on, to build intuition before any algebra. Watch out for: an open or filled dot at x=a does not change the limit. The limit from the left and the limit from the right must agree for the two-sided limit to exist. When they disagree, the limit does not exist.

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