(x - 0)
(x+h)^3-x^3=[(x+h)-x][(x+h)^2+ x(x+h)+h^2)
The original formula is equal to lim [(x+h) 3-x 3]/h.
(x - 0)
= lim [(x + h)^2+x(x+h)+h^2]
(x - 0)
=2h^2
2、lim = sin2x / sin7x
(x - 0)
Substitution formula of equivalent infinitesimal sinx ~ x (when x tends to 0)
The original formula is equal to
lim sin2x / sin7x
(x - 0)
= 2 times /7 times
=2/7
3、lim ( 1 - cos2x) / (xsinx)
(x - 0)
Substitution formula of equivalent infinitesimal (1) sinx ~ x (2)1-cosx ~ x 2/2 (when x tends to 0)
The original formula is equal to
lim ( 1 - cos2x) / (xsinx)
(x - 0)
=lim [(2x)^2]/2/x^2
=2
4.Lim 2 n * sin (x/2 n) (x is not equal to 0)
(n - ∞)
= Lin sin (x/2 n)
(n-∞) - * x
x/2^n
When n-∞, x/2 n (where x is regarded as a constant) tends to 0.
Substitution formula of equivalent infinitesimal sinx ~ x (when x tends to 0)
The original formula is equal to
=x
5、lim[ 1+2/x]^(x+3]
(x- infinity)
This type corresponds to 1∞ type, and the formula lim (1+1/x) (x) = e.
The original formula is equal to =
Forest [1+2/x] x * forest [1+2/x] 3
(x-∞) (x-∞)
=lim{[ 1+ 1/(x/2)]^(x/2)}^2 * 1
=e^2
6、lim[( 1 + x) / x]^2x
(x- infinity)
This type corresponds to 1∞ type, and the formula lim (1+1/x) (x) = e.
(x-∞)
Original formula = lim [( 1+ 1/x) x] 2.
(x- infinity
=e^2