Chapter 2: Limits

Question 1

Assume p & q are polynomials & a is a constant

1. Polynomials Functions: lim_x→ap(x) = p(a)
2. Rational Function: lim_x→ap(x)/q(x) = p(a) / q(a) (if q doesnt = 0)

Answer 1

Limits of Polynomial & Rational Functions (Theorem 2.4)

Question 2

Assume the function f(g(x)) & h satisfy f(x) greater than or equal to g(x) greater than or equal to h(x) for all values of x, ______ a, except possibly ____ a. If lim_x→af(x) = lim_x→a h(x) = L, then lim_x→ag(x) = L

Answer 2

Near a, Possibly at a (Squeeze Theorem)

Question 3

Suppose f is defined for all x near a. If f(x) grows arbitrarly large for all x sufficiently close (but not equal) to a, we write

lim_x→af(x) = inifinity

Answer 3

Inifinite Limits (Definition)

Question 4

If f(x) is negative & grows arbitrarly large in magnitude for all x sufficiently close (but equal) to a we write

lim_x→a f(x) = - infinity

Answer 4

Infinite Limts (Definition)

Question 5

If lim_x→a f(x) = + infinity, lim_x→a⁺ = + infinity,

lim_x→a^- = + infinity, the line x = a is called _______ of f

Answer 5

Verical Asymptote (Definition)

Question 6

If f(x) becomes arbitrarly large close to a finite number L for all sufficiently large & positive x, then we wrtie

lim_x→infinityf(x) = L, where y = L is a _________

Answer 6

Horizontal Asymptote (Limits at infinity & horizontal Asymptote definition)

Question 7

If m = n + 1 , then then lim_{x→<u>+</u> infinity} f(x) = infinity or - infinity & f has no horizontal asymptote but f has a ___________

Answer 7

Slant Asymptote

Question 8

Assuming f is in reduced form (p & q share no common factors), __________ occur at the zeros

Answer 8

Vaerical Asymptote

Question 9

The end behavior of e^x & e^-x (-infinity, +infinity) * ln x on (0, inifinity) is given by the following limits:

Answer 9

End Behavior of e^x,e^-x, & ln x

lim_x→Infinitye^x = infinity; lim_{x→-infinity}e^x=0

lim_x→infinitye^-x=0; lim_{x→-infinity} e^-x = infinity

lim_x→0⁺ln x= -infinity; lim_x→infinity ln x = infinity

Question 10

If f & g are continous at a. then the following functions are also continous at a. Assume c is a constant & n > 0 is an integer

Answer 10

Continuity Rules

1. f + g
2. f - g
3. cf
4. fg
5. f/g (if g isnt 0)
6. (f(x))ⁿ

Question 11

1. A polynomial function is continuous for all x
2. A rational function is continous for all x for which q doesnt = 0

Answer 11

Continuity Rules for polynomials & Rational functions

Question 12

If g is continous at a & f is continous at g(a) , then the composite function f(g(x)) is continous at a

Answer 12

Continuity of composite functions at a point

Question 13

1. If g is continous at a & f is continous at g(a) then,

lim_x→af(g(x)) = f (lim_x→ag(x))

2. If lim_x→ag(x) = L & f is continous at L then

lim_x→af(g(x)) = f (lim_x→ag(x))

Answer 13

Limits of Composite functions