Properties of Limits

1. Properties of Limits • There are several properties that allow limits to be evaluated analytically • Although it is not always true that lim f(x) = f(c) this is true for some functions • We can find the limit of these functions by direct substitution Theorem 1.1 If b and c are real numbers and n is a positive integer then • lim b = b lim 2 = 2 • lim x = c lim x = 5 • lim xn = cn lim x3 = (-2)3 = 8 x c x c x 5 x c x 5 x c x -2

2. f(x) g(x) Properties of Limits • The properties of limits stated in Theorem 1.1 apply to basic function operations Theorem 1.2 If b and c are real numbers, n is a positive integer, and f and g are functions such that lim f(x) = L and lim g(x) = K then • lim [b f(x)] = bL • lim [f(x) ± g(x)] = L ± K • lim [f(x) · g(x)] = L · K • lim = L/K provided that K ≠ 0 • lim [f(x)]n = Ln lim 3x2 = 3 · lim x2 = 3 · (lim x)2 = 3 · 4 2 = 48 x c x c x c x c x c x c x 4 x 4 x 4 x c

3. x2 + 2x + 1 x + 1 Properties of Limits • The properties in Theorem 1.2 can be applied to any polynomial function, and to any rational function with a non-zero denominator Theorem 1.3 If pis a polynomial function and cis a real number, then lim p(x) = p(c) If r is a rational function given by r(x) = p(x) / q(x) and cis a real number such that q(c) ≠ 0, then lim r(x) = r(c) = p(c) / q(c) • lim (3x2 + 2x – 1) = 3 · lim x2 + 2 · lim x – 1 = 3 · 42 + 2 · 4 – 1 = 55 • lim = (c2 + 2c + 1) / (c + 1), as long as c≠ -1 x c x c x 4 x 4 x 4 x c

4. c Properties of Limits • Expressions with radicals can be rewritten with rational exponents, giving the following theorem Theorem 1.4 If n is a positive integer, then lim = for any real number c when n is odd, or for any real number c > 0 when n is even • Direct substitution can also be used to find limits of compositions of functions if certain conditions are met Theorem 1.5 If f and g are functions such that lim g(x) = L and lim f(x) = f(L), then lim f [g(x)] = f [lim g(x)] = f (L) • Find lim • Since lim(5x2 + 1) = 81, and f(x) is an even root, lim = f(81) = 3 x c x c x L x c x c x 4 x 4 x 4

5. Properties of Limits • Direct substitution can be used to find the limits of trigonometric functions except at their asymptotes Theorem 1.6 If c is any real number in the domain of a given trigonometric function, then • lim sin x = sin c • lim cos x = cos c • lim tan x = tan c • lim cot x = cot c • lim sec x = sec c • lim csc x = csc c x c x c x c x c x c x c

6. x2 + 2x + 1 x + 1 Properties of Limits • If two functions agree at all but one point, and one of the functions has a limit at that point, then the other function has the same limit Theorem 1.7 If c is a real number and f(x) = g(x) for all x ≠c in an open interval containing c, and lim g(x) exists, then lim f(x) = lim g(x) • This theorem provides a way to find limits for functions with “holes” at certain values • For example, to find lim • Direct substitution gives the indeterminate form 0/0 • Divide out the common factor of x + 1 to get g(x) = x + 1 • The function g(x) is equivalent to f(x) except at x = -1 • Since lim g(x) = 0, lim f(x) is also zero x c x c x c x -1 x -1 x -1

7. x + 1 – 1 x(√x + 1 + 1) √x + 1 + 1 √x + 1 + 1 1 √x + 1 + 1 √x + 1 – 1 x √x + 1 – 1 x Properties of Limits • Another way to find equivalent functions when direct substitution yields an indeterminate form for the limit is to rationalize the numerator • For example, to find lim • Rationalize the numerator by multiplying by the conjugate g(x) = · = • Simplify the resulting expression to get g(x) = • Use direct substitution to get lim g(x) = ½ • Function g(x) is equivalent to f(x) except at x = -1 • Since lim g(x) = ½, lim f(x) is also ½ x 0 x 0 x 0 x 0

8. sin x x 1 – cos x x Properties of Limits • Some functions have limits that cannot be found directly • If two other functions that have limits surround the function of interest, all three functions may have the same limit Theorem 1.8 The Squeeze Theorem If h(x) ≤ f(x) ≤ g(x) for all x in an open interval containing c, except possibly at c itself, and if lim h(x) = L = lim g(x), then lim f(x) exists and is equal to L • The Squeeze Theorem can be used to find limits for two special trigonometric functions Theorem 1.9 Two Special Trigonometric Limits lim = 1 lim = 0 x c x c x c x 0 x 0