Usage of a great many, a great deal, a great number of

Usage of a great many, a great deal, a great number of

1. Add only countable plurals
a great many （of）
a good/great many books
a good/great many of my books
a number of=many
2. + uncountable noun
a great deal of
an amount of=amounts of
3. + uncountable noun / countable plural
a lot of
Please of
A large quantity of
Quantities of + uncountable noun, plural predicate
Which of a plenty of, a great deal of, a good many of, a number of adds an uncountable noun?
1 and 2
A great deal of plus uncountable, a plenty of countable and uncountable
A great deal of, a good many, a plenty of, a mumber of, a quantity of, quantities of
It's better to have an example
He has given me a great deal of help. A good many; a good many workers were for the bill.plenty Of (not having a) sufficient or considerable; countable
Given a > 0, set a = {X / - A-2
From the set B = {x power of X / a > 1},
When 0
Because the x power of a is greater than 1, so x > 0, that is, B = {X / x > 0}, a intersects B = empty set, so A-2 is less than or equal to 0, that is, a is less than or equal to 2, and - A-20, so 0 is obtained
The function f (x) defined in (- 1,1) satisfies 2F (x) - f (- x) = LG (x + 1), and the analytic expression of F (x) is obtained
This is the problem
Can we give a general way to solve this kind of problem
Because 2F (x) - f (- x) = LG (x + 1), --- (1) x is defined on (- 1,1), it is also true to replace x with - x, that is, 2f (- x) - f (x) = LG (1-x) -- (2) (2) left and right sides of formula (1) multiply by 2, then f (- x) can be eliminated, and 3f (x) = 2 * LG (1 + x) + LG (1-x)
By substituting - x into the equation, 2f (- x) - f (x) = LG (- x + 1), the system of equations can be obtained, and the solution is f (x) = (1 / 3) * (LG ((1-x) (1 + x) ^ 2))
In the end, how to judge whether the limit of a function exists? Every time we prove whether it is differentiable, we will use the definition. Finally, we will judge whether its limit exists. It is very fuzzy every time. Who can teach me?
The necessary and sufficient condition for the existence of limit is that the left limit and the right limit exist and are equal
The necessary and sufficient condition for derivability is that the left limit = the right limit and the limit value = the function value of F (x) at this point
So the limit must exist
Given a > 0, set a = {X / - A-2
From the set B = {x power of X / a > 1},
When 0
If (x 2-1) is a monotone function, then (x 2-1) = ∞ is a monotone function______ .
Let t = x2-ax-1, then y = LGT ∵ y = LGT is increasing in (0, + ∞), and ∵ function f (x) = LG (x2-ax-1) is monotone increasing in interval (1, + ∞), and ∵ t = x2-ax-1 is monotone increasing in interval (1, + ∞), and & nbsp; x2-ax-1 > 0 is constant in (1, + ∞), so A2 ≤ 1 and 1-a-1 ≥ 0 get a ≤ 0, so the answer is a ≤ 0
How to judge whether the limit of a function exists?
Let F: (a, + ∞) → R be a real valued function of one variable, a ∈ R. if for any given ε ＞ 0, there exists a positive number x, such that for all x suitable for inequality x ＞ x, the corresponding function value f (x) satisfies the inequality
F (x) = x power of 4 divided by (x power of 4 + 2), where x is a real number
Find f (1 / 2011) + F (2 / 2011) + F (3 / 2011) + +F (2009 / 2011) + F (2010 / 2011)
F (x) = 4 ^ X / (4 ^ x + 2) f (1-x) = 4 ^ (1-x) / [4 ^ (1-x) + 2] = 4 ^ (1-x) * 4 ^ X / 4 ^ x [4 ^ (1-x) + 2] = 4 ^ (1-x + x) / [4 ^ (1-x + x) + 2 * 4 ^ x] = 4 / (4 + 2 * 4 ^ x) = 2 / (4 ^ x + 2), so f (x) + F (1-x) = 4 ^ X / (4 ^ x + 2) + 2 / (4 ^ x + 2) = 1 Let s = f (1 / 2011) + F (2 / 2011) +... + F (2010 / 2011)