Limit of function When x tends to 2, y = x ^ 2 tends to 4 δ What is the absolute value of X-2

Limit of function When x tends to 2, y = x ^ 2 tends to 4 δ What is the absolute value of X-2

take δ= 0.0002, when 0 < |x-2|< δ When,
There are | y-4 | = | x + 2 | X-2 | = | X-2 + 4 | X-2 < (| X-2 | + 4 |) | X-2 | < 4.0002 * | X-2 | < 5 | X-2 | < 5 * 0.0002 = 0.001

Prove the function f (x) = - X by definition ²+ 2X + m is an increasing function on (- ∞, 1] Ask for detailed explanation

prove:
Optional x1

Prove that f (x) = 2x with the definition of monotonicity of function ²- 4X monotonically increases on (1, + ∞)

Take X1 and X2 at (1, + ∞) so that X1 > X2, then
f（x1）-f（x2）=2x1^2-4x1-2x2^2+4x2
=2(x1^2-x2^2)-4(x1-x2)
=2[(x1+x2)(x1-x2)]-4(x1-x2)
=2[(x1+x2)(x1-x2)-2(x1-x2)]
=2[(x1-x2)(x1+x2-2)]
∵x1＞x2,∴x1-x2＞0
∵x1,x2∈（1,＋∞）,∴x1＞x2＞1,∴x1+x2＞2,∴x1+x2-2＞0
∴[(x1-x2)(x1+x2-2)]＞0,∴2[(x1-x2)(x1+x2-2)]＞0
∴f（x1）-f（x2）＞0,f(x1)>f(x2)
∴f(x)=2x ²- 4X is an increasing function on (1, + ∞)
∴f(x)=2x ²- 4X monotonically increases on (1, + ∞)

Known function f (x) = 2x2-1 (1) Prove that f (x) is an even function by definition; (2) It is proved by definition that f (x) is a subtractive function on (- ∞, 0]; (3) Draw the image of function f (x), and write the maximum and minimum values of function f (x) when x ∈ [- 1,2]

(1) The domain of function f (x) = 2x2-1 is r
And f (- x) = 2 (- x) 2-1 = f (x)
The function f (x) is an even function;
(2) Proof: let x1 ＜ x2 ＜ 0,
Then f (x1) - f (x2) = 2x12-1 - (2x22-1) = 2 (x1 + x2) (x1-x2) > 0
∴f（x1）-f（x2）＞0
The function f (x) is a subtractive function on (- ∞, 0];
(3) Graph the function f (x)
The maximum and minimum values of function f (x) when x ∈ [- 1,2] are 7 and - 1, respectively

Prove the function f (x) = x + root sign (1 + x) by definition ²) It is an increasing function on R Solving speed

It is proved that let X1 and X2 be any two numbers in the definition field, and X1 > x2. F (x1) = X1 + root (square + 1 of x1) f (x2) = x2 + root (square + 1 of x2). Because X1 > X2, (square + 1 of x1) > (square + 1 of x2) so, (x1 + root (square + 1 of x1)) > (x2 + root (square + 1 of x2)) > (x2 + root (square + 1 of x2)), (x1

Known function f (x) = x / (x) ²- 1) , X belongs to (- 1,1). It is proved that f (x) is an odd function on (- 1,1) by definition 1) It is proved that f (x) is an odd function on (- 1,1) by definition 2) It is proved that f (x) is a subtractive function on (- 1,1) by definition 3) Solve the inequality f (m-1) + F (m) < 0 about M

1) The function definition domain (- 1, 1) is symmetric about the origin
f（-x）=-x/[(-x)^2-1]=-x/(x^2-1)=-f(x)
Therefore, f (x) is an odd function on (- 1,1)
2) For any x1, X2 ∈ (- 1,1), x1