Evaluating Limits Algebraically

Pre-Calculus · Axiom Academy

LESSON Evaluating Limits Algebraically When you can't just plug in: how factoring and conjugates rewrite a function to fill the gap. 1. First, Just Try Plugging In If a function has no break at the point you're approaching — and every polynomial is unbroken everywhere — the value it heads toward is exactly the value it takes there. So the first move is always the simplest one: substitute x = c and see what comes out. If f is continuous at c , the limit is just f(c) Watch the point ride the curve toward x = 3 . There's no jump, no hole — it lands cleanly, so the limit is the height it lands at. 2. When You Get 0/0: Factor and Cancel Sometimes substitution returns . That isn't zero and it isn't undefined — it's indeterminate , a signal that the top and bottom share a common factor that's forcing both to vanish. Factor them, cancel that shared factor, and the hole it created disappears. The graph of is just the line x+2 — with one point punched out at x = 2 , where the original is . The limit asks about the height near that point, and the line points straight at it. Watch the hole fill in as we cancel. Try substitution — it's indeterminate: Factor the numerator, then cancel the shared x-2 : Now substitute into the simplified form: 3. Square Roots: Rationalize with the Conjugate

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