F (x) = arctan1 / x, then the point where x is equal to zero is the jumping discontinuity. Why? When x tends to zero positive and X tends to zero negative, is there any difference in the limit of function

F (x) = arctan1 / x, then the point where x is equal to zero is the jumping discontinuity. Why? When x tends to zero positive and X tends to zero negative, is there any difference in the limit of function

When x = 0, 1 / X is meaningless, so it is a jumping point
Second, I don't know how to say, when it tends to 0 positive, 1 / X is infinite, when it tends to 0 negative, 1 / X is infinitesimal, and the corresponding f (x) value, i.e. limit, is also different

What kind of discontinuity of function f (x) = {X-1, x0 is x = 0?

lim(x→0-)f(x)=lim(x→0-)(x-1)=-1
LIM (x → 0 +) f (x) = LIM (x → 0 +) (x + 1) = 1
So LIM (x → 0 +) f (x) ≠ LIM (x → 0 -) f (x) ≠ f (0)
F (x) at x = 0 is the jump breakpoint of the first kind

Finding the discontinuous point of function f (x) = (1 + x) ^ [x / Tan (x - π)] in (0,2 π)
And judge its type

π / 2 and 3 π / 2 are the first kind of removable discontinuities (limits exist and are all 1)
π is the second kind of infinite discontinuity (x approaches infinity from positive direction, and 0 from negative direction)

Why is x = 0 the second kind of discontinuity of function f (x) = 1 / x?
For example, when x = 0, don't the left and right limits exist?

When x approaches + 0, the function f (x) = 1 / x approaches + ∞
When x approaches - 0, the function f (x) = 1 / x approaches - ∞
The left and right limits of the function do not exist, so it is the second kind of discontinuity

What's 999999 times 666

9999999x6666666
=(10000000-1)x6666666
=10000000x6666666-1x6666666
=66666660000000-6666666
=66666653333334

8888888 times 777 divided by 1111111 divided by 1111111?

(8888888 divided by 1111111) times (777 divided by 1111111)
=8×7=56

9999999×1+1111111×1=

10000000

A simple algorithm for 2222222 * 99999

2222222*(9999999+1-1)
2222222*10000000-2222222
=22,222,217,777,778

How simple is 156156x132132132 111111?

15615616x1321321 of 212121 11111 of 32132
=(156*1001001)/(21*10101) x (11*10101)/(132*1001001)
=156/21 x 11/132
=156/(21*12)
=13/21

Simple operation 36 * 111111 + 88888 * 8

36*111111+88888*8
=36*111111+111111*64
=111111*（36+64）
=111111*100
=11111100