Limits and Limit Theorems Quiz

Let f be a function defined at every number in some open interval containing a, except possibly at the number a itself. The limit of f(x) as x approaches a is L, written as L = lim (x -> a) f(x) if the following statement is true: Given any ε > 0, however small, there exists a δ > 0 such that 0 < |x - a| < δ implies |f(x) - L| < ε. Use the definition of a limit to prove: lim (x -> 2) (x^2 - 3) = 1.

Limit Theorem 1: Limit of a Linear Function. If m and b are any constants, prove that lim (x -> a) (mx + b) = ma + b.

Limit Theorem 2: Limit of a Constant. If c is a constant, then for any number a, prove that lim (x -> a) c = c.

Limit Theorem 3: Limit of the Identity Function. Prove that lim (x -> a) x = a.

Limit Theorem 4: Limit of the Sum and Difference of Two Functions. If lim (x -> a) f(x) = L and lim (x -> a) g(x) = M, then prove that lim (x -> a) (f(x) ± g(x)) = L ± M.

Limit Theorem 5: Limit of the Sum and Difference of n Functions. If lim (x -> a) f1(x) = L1, lim (x -> a) f2(x) = L2, ..., lim (x -> a) fn(x) = Ln, then prove that lim (x -> a) (f1(x) ± f2(x) ± ... ± fn(x)) = L1 ± L2 ± ... ± Ln.

Limit Theorem 6: Limit of the Product of Two Functions. If lim (x -> a) f(x) = L and lim (x -> a) g(x) = M, then prove that lim (x -> a) (f(x) * g(x)) = L * M.

Limit Theorem 7: Limit of the Product of n Functions. If lim (x -> a) f1(x) = L1, lim (x -> a) f2(x) = L2, ..., lim (x -> a) fn(x) = Ln, then prove that lim (x -> a) (f1(x) * f2(x) * ... * fn(x)) = L1 * L2 * ... * Ln.

Limit Theorem 8: Limit of the Quotient of Two Functions. If lim (x -> a) f(x) = L and lim (x -> a) g(x) = M, and M ≠ 0, then prove that lim (x -> a) (f(x) / g(x)) = L / M.

Limit Theorem 9: Limit of the nth Root of a Function. If lim (x -> a) f(x) = L, then prove that lim (x -> a) (f(x))^(1/n) = L^(1/n) if n is a positive integer.