Chapter 2- Limits

Question 0

Instantaneous rate of change

Answer 0

Limit of average rates of change

Question 1

Average rate of change

y = f(x) over an interval [x0,x1]

Answer 1

Average rate of change = delta f/delta x =
f(x1) - f(x0)
————–
x1 - x0

Question 2

Linear function

Answer 2

f(x) = mx + b

Question 3

Lim  f(x) = L
x-> c

The limit of f(x) as x approaches c is L

Answer 3

Either limit exists or limit does not exist

Question 4

Lim = L

x-> c-

Answer 4

If f(x) converges to L as x approaches c through values less than c.

Question 5

Lim = L

x-> c+

Answer 5

If f(x) converges to L as x approaches c through values greater than c.

Question 6

One sided limits

Answer 6

Limit only exists if both one sided limits exist and are equal

Question 7

Infinite limits

Answer 7

Limits can approach infinity or negative infinity

Question 8

Lim (f(x) + g(x)) =

x-> c

Answer 8

Lim f(x) + lim g(x) 
x-> c        x-> c

Question 9

Lim k f(x) =

x-> c

Answer 9

k lim f(x)

x-> c

Question 10

Lim f(x)*g(x) 
x-> c

Answer 10

Lim f(x) * lim g(x)
x-> c       x-> c

Question 11

f(x)
Lim ——- =
x-> c g(x)

Answer 11

lim f(x)
x-> c
---------
lim g(x)
x-> c

Question 12

Lim [f(x)]^(p/q) =

x-> c

Answer 12

(lim f(x))^(p/q)

x-> c

Question 13

Assume that f(c) is defined on an open interval containing x = c.

Answer 13

Then f is continuous at x = c if

lim f(x) = f(c)
x-> c

Question 14

If limit does not exist or if it exists but is not equal to f(c)

Answer 14

We say that f has a discontinuity

Question 15

For a function to be continuous

Answer 15

1. f(c) is defined.
2. lim f(x) exists.
   x-> c
3. They are equal.

Question 16

Removable discontinuity

Answer 16

If lim f(x) exists but is not equal to f(c)

x-> c

Question 17

Jump discontinuity

Answer 17

Occurs if the one-sided limits lim x-> c- f(x) and lim x-> c+ f(x) exist but aren’t equal

Question 18

Infinite discontinuity

Answer 18

If one or both of the one sided limits is infinite

Question 19

Substitution method

Answer 19

If f(x) is known to be continuous at x = c, then the value of the 
lim f(x) is f(c).
x-> c

Question 20

Indeterminate forms

Answer 20

0/0
Infinity/infinity
Infinity*0
Infinity - infinity

Question 21

If f(x) is indeterminate at x = c

Answer 21

Transform f(x) algebraic ally into a new expression that is defined and continuous at x = c. Then evaluate by substitution.

Question 22

sin x
Lim ——— =
x-> 0 x

Answer 22

1

Question 23

1 - cos x
Lim ————- =
x-> 0 x

Answer 23

0

Question 24

Squeeze Theorem

Answer 24

We say that f(x) is squeezed at x = c if there exists functions l(x) and u(x) such that 
l(x) < f(x) < u(x) for all x not equal to c in an open interval I containing c, and
Lim f(x) = L
x-> c

Question 25

Lim x^n =

x-> infinity

Answer 25

Infinity

Question 26

Lim x^-n =

x-> -infinity

Answer 26

0

Question 27

3x^4
Lim ——- =
x-> c 7x^4

Answer 27

3/7

Question 28

3x^3
Lim ———- =
x-> infinity 7x^4

Answer 28

0

Question 29

3x^8
Lim ———- =
x-> -infinity 7x^3

Answer 29

-infinity

Question 30

3x^7
Lim ———- =
x-> -infinity 7x^3

Answer 30

Infinity

Question 31

Intermediate Value Theorem

Answer 31

Continuous function cannot skip values

If f(x) is continuous on [a,b] with f(a) not equal to f(b), and if M is a number between f(a) and f(b), then f(c) = M for some c E (a,b).

Question 32

Formal definition of a limit

Answer 32

Learn it with problems in notebook