Indeterminate Forms and Algebraic Techniques

Calculus 1 · Axiom Academy

LESSON Indeterminate Forms & Algebraic Techniques When direct substitution gives , the limit isn't gone — algebra clears the form and reveals it. 1. The Form — Factor and Cancel Take . Substituting x=2 makes the numerator and denominator both 0 — the dreaded . It is called indeterminate because has no single value on its own; the limit may still exist, but substitution can't see it. The shared factor (x-2) cancels, and now x=2 slots in: 2+2=4 . 2. A Square-Root Difference — Multiply by the Conjugate Now . Substituting x=0 gives again — but factoring won't touch that square root. Instead multiply top and bottom by the conjugate . The product (a-b)(a+b)=a^2-b^2 kills the radical: the numerator becomes (x+1)-1=x . 3. The Form — Divide by the Highest Power Limits at infinity bring a new form. Consider . As x grows, the top and bottom both run off to — the indeterminate . Divide every term by the highest power present , x^2 : each and term decays to 0 , and only the leading coefficients survive. Every term vanishes, leaving — the ratio of leading coefficients. 4. The Form — Combine Into One Fraction Last form: . As each piece blows up, and we're subtracting — which could be anything. The fix is to combine into a single fraction over the common denominator x^2-1=(x-1)(x+1) . The two infinities merge, a factor of (x-1) appears, and it cancels. The numerator (x+1)-2=x-1 cancels the denominator's (x-1) , leaving .

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