Find the limit of the function cos (π / x) / (2 - √ (2x)) when x tends to 2 Can you give me a specific process... And what is the Robida rule? It seems that it hasn't been taught yet

Find the limit of the function cos (π / x) / (2 - √ (2x)) when x tends to 2 Can you give me a specific process... And what is the Robida rule? It seems that it hasn't been taught yet

lim(x->2) cos(π/x)/(2-√(2x))
=lim(x->2) sin(π/2 - π/x)/(2-√(2x))
∵ X - > 2, (π / 2 - π / x) - > 0, Equivalent Infinitesimal Substitution:
sin(π/2 - π/x) ~ (π/2 - π/x)
=lim(x->2) (π/2 - π/x)/(2-√(2x))
Multiplication of numerator and denominator [2 + √ (2x)] rationalization
= π*lim(x->2) [(x-2)/2x] [2+√(2x)]/(4-2x)
= π*lim(x->2) (x-2)[2+√(2x)]/[2x*2(2-x)]
= -π*lim(x->2) [2+√(2x)]/[2x*2]
= -π/2

Bounded function

Many people found in Baidu forget the boundedness of function. The definition of boundedness of function: let the definition field of function f (x) be D and the number set X ⊆ D. if there is a number K1 such that f (x) ≤ K1 is true for any x ∈ x, then function f (x) has an upper bound on X. and K1 is called an upper bound of function f (x) on X. In addition, if there is a number

Asking a high number question is related to the boundedness of the function It is proved that arctanx / 1 + x ^ 2 is a bounded function No idea at all

0 ≤ |arctanx /(1+X^2) | = |arctanx| *1/(1+X^2) ≤ |arctanx| ≤π/2
‡ arctanx / 1 + x ^ 2 is a bounded function on R, the upper bound can be π / 2 and the lower bound can be - π / 2

Boundedness of higher number functions Function, boundedness of sequence. The book stipulates that | f (x) | less than or equal to m is bounded. If - 3 is less than or equal to | g (x) | less than or equal to 2, - 3 and 2 are not the same number, does the boundedness of G (x) mean that | g (x) | less than or equal to 3?

Boundedness of functions
Definition: in a process, there is a variable y. if there is a positive number a, a time can be found in the process
After this moment, there will always be ∣ y ∣ 10, and there will always be 1 / 2 ⁿ

A problem of judging whether a function is bounded in a high number Prove that the function f (x) = x / (x) ²+ 1) Bounded on R (in the following symbols [] represents absolute value!) （1-X) ² ≥ 0, so [1 + x] ²] ≥ 2 [x], so [f (x)] = [x / (x) ²+ 1)]=2[X]/[1+X ²] ≤1/2 I want to ask [1 + x] ²] How is ≥ 2 [x] obtained?

Mean inequality a ^ 2 + B ^ 2 > = 2Ab
Think of x ^ 2 as the square of [x]

High number function continuity exercises Discuss the function f (x) = 2x, 0 ≤ x ≤ 1, 3-x, 1

The definition of continuity is that the value of a function is equal to the limit
First, calculate whether the lower limit exists
Left limit = 2
Right limit = 2
The left and right limits are equal, so the limit exists and is 2
Let's look at the function value
f(1)=2
Then the value of the function is equal to the limit
Therefore, f (x) is continuous at x = 1