quiz 4

Question 1

function f(x) is continuous at x = a if all 3 conditions hold

Answer 1

1. f(a) is defined.
2. lim as x approaches a of f(x) exists (as a finite number).
3. lim as x approaches a of f(x) = f(a).

Question 2

where are vertical asymptotes found

Answer 2

denominator is 0 but numerator is not 0

Question 3

how to find the limit manually of rational functions

Answer 3

pick values that are REALLY close to the denominator being undefined and compare their y values

• if they’re changing drastically, it’s heading to some form of infinity

Question 4

shortcut for doing c/0 limits

Answer 4

box trick

• pick a value from the left and from the right that would make denominator undefined and make a box filling in - or + value for both numerator and denominator.
• if being asked for limit approaching from left, only choose value from left side and same w right
• if asked limit, choose from both sides

one box for each factor

Question 5

any time you have to sub 0 for ln(x), answer is always

Answer 5

heading to negative ∞

Question 6

how to make the domain all real numbers if you draw an asymptote?

Answer 6

put a dot on it

Question 7

what would the graph look like for the following:

limx→3 f(x) exists but f(3) does not exist

Answer 7

• open circle
• same limit from left and right

Question 8

what would the graph look like for the following:

limx→3 f(x) does not exist and f(3) does not exist.

Answer 8

“break”

• 2 open circles at different points at the same x value

Question 9

what would the graph look like for the following:

both f(3) and limx→3 f(x) exists, but limx→3 f(x) does not equal f(3)

Answer 9

“hole with a dot”

• both limits approaching the open circle, while dot is below or above open circle

Question 10

how to find limit of a linear function algebraically?

Answer 10

just sub in the x value

Question 11

3 functions whose limit can be evaluated algebraically

Answer 11

1. polynomials
2. rational functions (continuous at every x in their domain)
3. exponential, logarithmic, and trig functions at their domain

Question 12

limx→4 √x-4

why is this a special case

Answer 12

y value when sub in is 0 but that is not the answer b/c there is no left sided limit

for limit to exist, need to come in from both sides

correct answer is DNE

Question 13

what happens in the 0/0 case for limits?

Answer 13

1. factor
2. cancel
3. substitute

• if its still undefined, becomes a c/0 case and the approach infinity

Question 14

multiplying by rational conjugate in limits

Answer 14

just change the sign b/w the radical # and sum and multiply both top and bottom