At the beginning  F(x) = sin(x)/x  is calculated.

function F(x) { with(Math) {
y = sin(x)/x
return y
}}

For example, I can deduce that, as x → 0, sin(x)/x → 1:

and that, as x → ∞, sin(x)/x → 0:

 

(tan(x)-sin(x)+1-cos(x)) / (x*x+1-cos(x))  as  x → 0

function F(x) { with(Math) {
y = (tan(x)-sin(x)+1-cos(x)) / (x*x+1-cos(x))
return y
}}

The limit is 1/3.

√(x*x+x)-√(x*x-2)  as  x → ∞

function F(n) {
with(Math) {
y = sqrt(x*x+x)-sqrt(x*x-2)
return y
}}

The limit is 1/2.

n / (n!)1/n  as  n → ∞

function F(x) { with(Math) {
 y = x / pow(Fat(x), 1/x)
 return y
}}

function Fat(n) { with(Math) {
 y = 1; for (k=1; k <= n; k++) {y = y*k}
 return y
}}
 [170,  171,  172,  173,  174,  175 ->]
 2.663087878748024, 0, 0, 0, 0, 0
   --------
 [150,  160,  170,  180,  190 ->]
 2.6569209686814057, 2.660179125103347, 2.663087878748024, 0, 0
   --------
 [1,  50,  100,  150,  200 ->]
 1, 2.5663063992029507, 2.6320853228463696, 2.6569209686814057, 0

171 → 0  because (for computer)  171! = infinity  (170! = 7.257415615307994e+306).  F(170) = 2.663087878748024.  It can be proven that (as N → ∞) F(N) → e = 2.71828182...

F(x) = sin(x)−tan(x) as x → 0.  Limit is 0. With which order F(x) tends to 0?

function F(n) {
with(Math) {
y = sin(x)-tan(x)
return y
}}
 [1e-1,  1e-2,  1e-3,  1e-4,  1e-5  -> ]
 -0.0005012554386223944, -5.00012500542768e-7, -5.000001250705932e-10, -4.999999980020986e-13, -5.000001606324245e-16

If x is divided by 10, F(x) is divided by 1000. F(x) is an infinitesimal of order 3.

sin(x)-tan(x)x3Q = ?

I study the limit of (sin(x)-tan(x))/x³:

function F(x) { with(Math) {
 y = (sin(x)-tan(x))/pow(x,3)
 return y
}}

 [1e-1, 1e-2, 1e-3, 1e-4, 1e-5 ->]
 -0.5012554386223943, -0.5000125005427679, -0.5000001250705932, -0.4999999980020986, -0.5000001606324244

I can deduce that Q = −1/2, that is, that (as x → 0) sin(x)-tan(x)−1/2·x3.

F(x) = log(1+x2)/(sin(xk). How much is the limit for x which tends to 0 when k changes?

function F(x) { with(Math) {
 k = 2
 y = log(1+x*x)/sin(pow(x,k))
 return y
}}

[k = 4:  1e-2, 1e-3, 1e-4, -1e-2, -1e-3, -1e-4 ->]
9999.500033329732, 999999.4999180668, 99999998.8922529, 9999.500033329732, 999999.4999180668, 99999998.8922529
  --------
[k = 3:  1e-2, 1e-3, 1e-4, -1e-2, -1e-3, -1e-4 ->]
99.99500033331397, 999.9994999180667, 9999.999889225292, -99.99500033331397, -999.9994999180667, -9999.999889225292
[k = 0:  1e-2, 1e-3, 1e-4, -1e-2, -1e-3, -1e-4 ->]
0.00011883356899837213, 0.0000011883945114831993, 1.1883950926137091e-8, 0.00011883356899837213, 0.0000011883945114831993, 1.1883950926137091e-8
  --------
[k = 1:  1e-2, 1e-3, 1e-4, -1e-2, -1e-3, -1e-4 ->]
0.00999966669360799, 0.0009999996665846695, 0.00009999999905891957, -0.00999966669360799, -0.0009999996665846695, -0.00009999999905891957
  --------
[k = 2:  1e-2, 1e-3, 1e-4, -1e-2, -1e-3, -1e-4 ->]
0.9999500049995564, 0.9999994999182334, 0.9999999889225291, 0.9999500049995564, 0.9999994999182334, 0.9999999889225291

k = 2, F(x) → 1.   k = 1, F(x) → 0.   k = 0, F(x) → 0.
k = 3, 5, ...; if x → 0+, F(x) → ∞; if x → 0-, F(x) → −∞.
k = 4, 6, ..., F(x) → ∞.