Mastering the Fundamentals: Important Limit Properties in Calculus for Evaluating Functions

Limit Properties

Limit of b f(x)=b LLimit of f(x)+/-g(x)= The limit of the two combined/subtractedLimit of fg= LK (K is the limit of g(x))Limit of f(x)/g(x)=L/KLimit of f(x)^n=L^n

Limit properties are fundamental rules of calculus that help simplify the process of evaluating limits of functions. Here are some of the most important limit properties:

1. Sum and Difference Properties: If lim(f(x)) = L and lim(g(x)) = M, then
lim[f(x) ± g(x)] = L ± M.

2. Product Property: If lim(f(x)) = L and lim(g(x)) = M, then
lim[f(x) * g(x)] = L * M.

3. Quotient Property: If lim(f(x)) = L and lim(g(x)) = M (with M ≠ 0), then
lim[f(x) / g(x)] = L / M.

4. Power Property: If lim(f(x)) = L, then for any positive integer n,
lim[f(x)^n] = L^n.

5. Root Property: If lim(f(x)) = L (with L > 0) and n is a positive integer, then
lim[nth-root(f(x))] = nth-root(L).

6. Trigonometric Properties: If lim(x → 0) [sin x / x] = 1, then:
lim(x → 0) [cos x – 1] / x] = 0,
lim(x → 0) [(1 + x)^a – 1] / x] = a,
lim(x → 0) [(e^x – 1) / x] = 1.

These limit properties provide a systematic approach for evaluating limits of functions, and they form the basis for more advanced calculus concepts, such as derivatives and integrals.

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