Study Flashcards
(48 cards)
Why is lim sin3x / x = 0?
x—> ∞
Because lim sinx / x = 0,
x—> ∞
So multiply top and bottom by 3 and cancel.
How do you find the limit at infinity for lim 4x^2 + x + 1 / 2x^3 - 5?
x—> ∞
Divide everything by the highest power of x, simplify, then replace x with ∞, which is same as saying 4/0 so they all become zero.
Do this example:
lim 6-√x / 36x-x^2
x—>36
1/432
Do this example:
lim t^2-4 / 2t^2 + 5t+2
x—>-2
4/3
How is the the graph of a limit if lim f(x)=-1, lim f(x)=2, f(0)=1
x—>0- x—>0+
Open circle at y=2 and line to the right of it, closed circle at y=1, and open circle at y=-1 with line to the left of it
If you are to find the limit coming from the left or right side on a graph, what is done?
Use the closet line to the point, and the first y-value is the limit.
Use the closet line to the point, and the first y-value is the limit.
If you are to find the limit coming from the left or right side on a graph
What does this mean?:
lim f(x) = -∞ x—>3+
The values of f(x) can be made negative with arbitrarily large absolute values by taking x sufficiently close to 3.
The values of f(x) can be made negative with arbitrarily large absolute values by taking x sufficiently close to 3.
lim f(x) = -∞ x—>3+
What does this mean?:
lim f(x) = ∞ x—>-4
The values of f(x) can be made arbitrarily large by taking x sufficiently close to -4.
The values of f(x) can be made arbitrarily large by taking x sufficiently close to -4.
lim f(x) = ∞ x—>-4
Do this example:
Find continuity and domain for:
f(x)=ln(x)+tan^-1(x) / x^2 -1
ln(x) continuous on (0, ∞)
tan^-1(x) continuous for (-∞, ∞)
x^2 -1 continuous for (-∞, ∞)
f(x) continuous on {A∩B∩C | x^2 -1 ≠ 0}
Domain: (0,1)U(1, ∞)
List all continuous types of functions.
Polynomials, trigonometric, exponential, rational, inverse trigonometric, logarithmic, root.
Polynomials, trigonometric, exponential, rational, inverse trigonometric, logarithmic, root.
All continuous types of functions.
If f and g are continuous at a, and c is a constant, then which functions are also continuous?
f+g, f-g, cf, fg, f/g for g≠0. Can be proved with limit laws and hold on intervals.
f+g, f-g, cf, fg, f/g for g≠0. Can be proved with limit laws and hold on intervals.
If f and g are continuous at a, and c is a constant, then these functions are also continuous.
Do this example:
lim x^3 • cos(1/x)
x—>0
lim x^3 • cos(1/x)= 0
x—>0
Do this example:
lim xsin(x) x—>0
lim xsin(x)=0 x—>0
Do this example:
5x+2<=f(x)<=x^2 + 8 as x—>2
lim f(x) = 12 x—>2
Do this example:
Where is this function continuous?:
f(x)=x^2 -3x-10 / x^2 -8x+15
f(x) = {x+2 / x-3 if x≠5,3 DNE if x=5,3}
How do you check if
f(x)=1-√1-x^2 is continuous on
x—>a
the interval [-1,1]?
Check if check if f(x) = f(a)
Check if check if f(x) = f(a)
When checking if continuous on an interval or a point.
lim f(x) and lim f(x) = what? x—>a- x—>a+
f(a)
Draw the graph of [|x|], then determine when it is discontinuous.
Discontinuous when approaching from left, as it is a different y-value.
