Limits Flashcards
Any limit divided by 0 is…
Undefined
A circle (gap) in a function means that it is…
Undefined at that point
To find a limit at a gap in a function…
Look at what the function is approaching. What is the limit as we get closer to our undefined point?
If a function graph has 2 points on the same x level, the function is…
Undefined at that point
To define a function with 2 points on the same x level…
Approach the point from the left and right side.
In lim X —> a-, “-“means…
Approaching from the left (smaller numbers)
In lim X —> a+, “+“means…
Approaching from the right (bigger numbers)
lim f(x) + g(x) =
lim f(x) + lim g(x)
lim f(x) - g(x) =
lim f(x) - lim g(x)
lim f(x) * g(x) =
lim f(x) * lim g(x)
lim f(x) ÷ g(x) =
lim f(x) ÷ lim g(x)
If f(x) is a continuous function for all real numbers, then
lim f(x) = ?
x—>c
f(c)
(Plug in c for x)
If f(x) = p(x) / q(x), and q(x) = 0, then
lim f(x) does not exist OR
x=c is a vertical asymptote (exponential function to ∞)
To find the limit of lim f(x) / g(x)
x—> ∞
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)
lim sin ax / ax =
x—>0
1
lim ax / sin ax =
x—>0
1
lim 1 - cos ax / ax =
x—>0
0
If f(x) is continuous at x=b, and lim g(x) = b, then
x—>a
lim f(g(x)) =
x—>a
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)
When do we use L’Hopital’s rule
When plugging in values results in:
0/0, ∞/∞, ∞−∞, 0*∞, 0⁰, 1^∞
What is L’Hopital’s Rule
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
What do we use if L’Hopital’s rule does not work?
Simplify the expressions, then plug in the value again
lim sin ax / ax =
1
