AP Calc BC Unit 1 Concepts

3 reasons why limits fail to exist

1. f(x) approaches a different number from the right side of c than it approaches from the left

2. f(x) increases or decreases without bound as x approaches c (there is an asymptote).

3. f(x) oscillates between two fixed values as x approaches c

3 criteria for continuity

For a function to be continuous at a point x=c…

1. f(c ) must be defined

2. lim f(x) as x—>c exists

3. lim f(x) as x—>c = f(c )

Removeable discontinuity

A hole in the graph caused by a common factor in the numerator and denominator canceling out

Non-removable discontinuity

Jump discontinuities and essential discontinuities (asymptotes)

Properties of limits

1. lim [bf(x)] as x—>c = bL

2. lim [f(x) ± g(x)] as x—>c = L ± K

3. lim [f(x)*g(x)] as x—>c = L*K

4. lim [f(x)/g(x)] as x—>c = L/K, K can’t equal 0

5. lim [f(x)]^m/n as x—>c = L_^m/n

Rule for limits of composite functions

lim f(g(x) as x—>c = f(lim g(x) as x—>c) provided that lim g(x) as =K and f(x) is continuous at K

What do you do when a limit is equal to #/#?

basically lim f(x) as x—>x=f(c )

Direct Substitution

What do you do when the limit equals 0/0 (DNE)?

This means there is a hole at x=c, so you should factor, use the common denominator method, or multiple by a conjugate.

What do you do when the limit equals #/0 (DNE)?

This means there is a VA at x=c, so check the limit from the right and the left to see if they approach the same value or not.

Squeeze Theorem

If h(x)=/<f(x)=/<g(x) for all of x in an open interval containing c, and f is continuous, then lim h(x) as x—>c = K = lim g(x) as x—>c, which means that lim f(x) as x—>c = K also.

Limit At Infinity Theorem

if r>0 as a rational #, then lim (1/x^r) as x—>∞ = 0
If r>0 is a ration # such that x^r is defined for all x, then lim (1/x^r) as x—> -∞ = 0

Finding Limits at ±∞ of Rational Functions P(x)/Q(x)

1. If the degree of the numerator is = to the degree of the denominator, lim f(x) as x—>±∞ = LC of num/LC of den

2. If the degree of the numerator is > than the degree of the denominator, the limit of the function DNE

3. If the degree of the numerator is < than the degree of the denominator, the limit of the function = 0

The Intermediate Value Theorem

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

Rule for lim k/x as x—>±∞

if k is a positive constant, then lim k/x as x—> 0+ = ∞ and lim k/x as x—> 0- = -∞ (making this limit nonexistant)

Rule for lim k/x² as x—>0

if k is a positive constant, then lim k/x² as x—> 0+ = ∞ and lim k/x² as x—> 0- = ∞, making lim k/x² as x—> 0 = ∞

lim sin(x)/x as x—>0 = __

1

lim sin(x)/cos(x) as x—>0 = lim tan(x) = __

0 (because tan(0) = 0)

lim sin(x) as x—>c = __

sin(c )

lim cos(x) - 1/x as x—>0 = __

0

lim sinax/x as x—>0 = __

lim sin4x/x as x—>0 = __

a

4