Computing Limits Algebraically

Business Calculus · Axiom Academy

LESSON Computing Limits Algebraically Techniques for handling indeterminate forms like 0/0 When Direct Substitution Fails You learned that for most functions, you can compute limits by plugging in the value. But sometimes this gives 0 , which is indeterminate —it doesn't tell us the answer. 0 is NOT equal to 0, 1, or any other number. It's indeterminate —the limit could be anything! We need to do more work to find the actual value. The good news: when you get 0 , the limit usually does exist. You just need algebraic techniques to find it. The most common technique is to factor the numerator and denominator, then cancel the common factor causing the 0/0. When both numerator and denominator are zero at x = a, they share a common factor of (x - a). Canceling this factor removes the 0/0 problem! When square roots are involved, multiply by the conjugate to eliminate the radical. Direct substitution: - 2 0 = 0 Multiplying gives: ( - b)( + b) = a - b^2 (no more square root!) Technique 3: Combining Fractions When you have complex fractions, combine them into a single fraction first. This type of limit appears when computing derivatives using the definition. The expression above is actually the derivative of f(x) = x+2 at x = 0! Some limits appear so frequently that it's worth memorizing them: Business Application: Continuous Growth Rate The limit _ x 0 x = 1 tells us that for small growth rates, e^r - 1 r .

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