Small value approximation

Try prove the following:

1. ln(1+x) ~ x

2. (1+bc)^a -1 ~ abc

3. arcsin(x) ~ x

4. arctan(x) ~ x

Proof for 1:

Recall the power series for e^x, now we have:

1+x < e^x

ln(1+x) < x

Take the limit when x approaches zero, QED.

For 2, use L'Hospital Rule:

For 3:

arcsin(x) ~ x because

x ~ sin(x)

Likewise, for 4:

arctan(x) ~ x because

x ~ tan (x)

Note that for |x| < pi/2:

sin(x) < x < tan(x)

QED

5. cos(x) ~ 1 - x^2/2

Note: both cosine and polynomials are continuous everywhere and differentiable, so we can compute the derivative from both sides:

-sin(x) ~ 0 - 2x/2x

Which is equivalent to:

sin(x) ~ x

QED

Note: Don't differentiate it further:

cos(x) ~ 1

The above approximation is invalid because the variables must be present on both sides.