calculus

Verify the given linear approximation at $a=0$. Then determine the values of x for which the linear approximation is accurate within 0.1: $$ln(1 + x) ≈ x $$

The official method that was used to calculate a question similar to this was to use this formula:

$$ |f(x) - L(x)| < 0.1 $$

for example, in this case, it would be:

$$|ln(1 + x) - x| < 0.1 $$

My first question is, how do you solve for x algebraically in the equation above?

My second question is why isn't this the equation?

$$ |f'(x) - L(x)| < 0.1 $$

Logically I think the equation above makes more sense because you want to compare the approximation of the tangent line (L(x)) with the actual tangent line (f'x).