Limit Laws

Business Calculus · Axiom Academy

Rules for computing limits algebraically So far, you've learned to find limits by looking at graphs. But in practice, we need to compute limits algebraically —without drawing pictures. The limit laws tell us how limits interact with arithmetic operations. They're based on a simple intuition: If two functions are approaching certain values, then their sum, difference, product, and quotient also approach predictable values. All limit laws assume that the individual limits exist and are finite . If _ x a f(x) = L and _ x a g(x) = M , then we can combine them. Given that _ x a f(x) = L and _ x a g(x) = M : The limit of a sum equals the sum of the limits. The limit of a difference equals the difference of the limits. Constants can be pulled out of limits. The limit of a product equals the product of the limits. The limit of a quotient equals the quotient of the limits (if the denominator isn't zero). For any positive integer n, the limit of a power equals the power of the limit. For positive integer n, the limit of a root equals the root of the limit (assuming the root exists). Answer: _ x 2 (3x - 1)^4 = 625 The Direct Substitution Property For most "nice" functions (polynomials, rational functions at non-zero denominators, exponentials, logs), there's an easy method: If f(x) is a polynomial, rational, exponential, or logarithmic function and f(a) is defined, then: Find _ x 3 (x^3 - 2x^2 + 5x - 1) When Direct Substitution Fails

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