Calc BC Test 1

Limit Definition

A limit is a number (L) that the y-values of a function f(x) can be made arbitrary close to (within any small error e) by restricting x-values to some neighborhood (within distance delta of point)

Limit of constant

lim(k) as x → a = k

limit with constant coefficient

lim(cf(x)) as x → a = clim(f(x)) as x → a

limit of sum and difference

lim(f(x)+-g(x)) as x → a = lim(f(x)) as x → a +- lim(g(x)) as x → a

limit of product

lim(f(x)×g(x)) as x → a = lim(f(x)) as x → a × lim(g(x)) as x → a

limit of quotient

lim(f(x)/g(x)) as x → a = lim(f(x)) as x → a /lim(g(x)) as x → a, limg(x) isn’t 0

Limit of composite function

If limg(x) as x → c = L and limf(x) as x → L = f(L), then limf(g(x)) as x → c = f(limg(x) as x → c) = f(L)

Infinite Limits

If as x → c, the function does not approach anything, but rather continues infinitely in the positve or negative direction, the limit is a special DNE case where it equals infinity.

Vertical asymptote

If f(x) approaches infinity (or negative infinity) as x → c, then we call x = c a vertical asymptote of the graph of f. If the limit DNE but the left or right do not approach infinity, it is an empty point.

Limits at infinity

The limit of any rational function as x → infinity is equal to the limit of the numerator and denominator leading term as x → infinity.

Horizontal Asymptote

The line y = b is a horizontal asymptote of function y = f(x) if limf(x) as x → infinity = b or imf(x) as x → negative infinity = b

Continuity at a point

f(x) is continuous at x = c iff:

1. f(c) is defined

2. limf(c) as x → c exists and is finite

3. f(c) = limf(c) as x → c

Continuity on open interval

f(x) is continuous on an open interval (a,b) if it is continuous at each point inside (a,b)

continuity on closed interval

A function is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and limf(x) as x→a+ = f(a) and limf(x) as x→b+ = f(b)

The Intermediate Value Theorem: Theorem 1

If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), inclusive, then there is at least one number c in the interval [a,b] such that f(c) = k

Between two given y-values continuous fn goes through every y-value on the interval

The Intermediate Value Theorem: Theorem 2

If f is continuous on the closed interval [a,b] continuous on the closed interval [a,b] and f(a) and f(b) are nonzero and have opposite signs, then there is at least one solution of the equation f(x) = 0 in the interval (a,b)