Limits

Question 1

Any limit divided by 0 is…

Answer 1

Undefined

Question 2

A circle (gap) in a function means that it is…

Answer 2

Undefined at that point

Question 3

To find a limit at a gap in a function…

Answer 3

Look at what the function is approaching. What is the limit as we get closer to our undefined point?

Question 4

If a function graph has 2 points on the same x level, the function is…

Answer 4

Undefined at that point

Question 5

To define a function with 2 points on the same x level…

Answer 5

Approach the point from the left and right side.

Question 6

In lim X —> a-, “-“means…

Answer 6

Approaching from the left (smaller numbers)

Question 7

In lim X —> a+, “+“means…

Answer 7

Approaching from the right (bigger numbers)

Question 8

lim f(x) + g(x) =

Answer 8

lim f(x) + lim g(x)

Question 9

lim f(x) - g(x) =

Answer 9

lim f(x) - lim g(x)

Question 10

lim f(x) * g(x) =

Answer 10

lim f(x) * lim g(x)

Question 11

lim f(x) ÷ g(x) =

Answer 11

lim f(x) ÷ lim g(x)

Question 12

If f(x) is a continuous function for all real numbers, then
lim f(x) = ?
x—>c

Answer 12

f(c)
(Plug in c for x)

Question 13

If f(x) = p(x) / q(x), and q(x) = 0, then

Answer 13

lim f(x) does not exist OR
x=c is a vertical asymptote (exponential function to ∞)

Question 14

To find the limit of lim f(x) / g(x)
x—> ∞

Answer 14

Plug in infinity, and remove anything that is not the highest power of ∞.

If the bottom is bigger, lim = 0
If the top is bigger, lim = ∞
If top and bottom are equal, lim = c (use the coefficients)

Question 15

lim sin ax / ax =
x—>0

Answer 15

1

Question 16

lim ax / sin ax =
x—>0

Answer 16

1

Question 17

lim 1 - cos ax / ax =
x—>0

Answer 17

0

Question 18

If f(x) is continuous at x=b, and lim g(x) = b, then
x—>a
lim f(g(x)) =
x—>a

Answer 18

f(lim g(x)) = f(b)
x—>a

(Find limit if inside function, plug it in (substitute each “x” in f(x) with inner function limit) and find outside limit)

Question 19

When do we use L’Hopital’s rule

Answer 19

When plugging in values results in:

0/0, ∞/∞, ∞−∞, 0*∞, 0⁰, 1^∞

Question 20

What is L’Hopital’s Rule

Answer 20

When direct substitution results in an indeterminate form,
differentiate the top and bottom, then plug in values again

lim p(x) / q(x) = lim p’(x) / q’(x)
x—>c x—>c

Question 21

What do we use if L’Hopital’s rule does not work?

Answer 21

Simplify the expressions, then plug in the value again

Question 22

lim sin ax / ax =

Answer 22

1