I've some trouble understanding how to work with the natural logarithm when trying to calculate a limit. I just can't seem to fit what I know together:

- I know that if $\displaystyle x \to 1$, then $\displaystyle \ln x \approx x - 1$, because $\displaystyle \ln(1) = 0$, as $\displaystyle e^0 = 1$. $\displaystyle \ln x, 0 < x < 1$ is a negative, so is $\displaystyle x - 1, x < 1$, so their quotient is a positive.

- I know the given standard limits: $\displaystyle \lim_{x \to 1}\frac{\ln x}{x -1} = 1$ (following from the above: $\displaystyle \frac{a}{a} = 1$) and $\displaystyle \lim_{x \to 0}\frac{\ln (1 + x)}{x } = 1$

But, I just can't seem to fit it together when I'm asked to solve an exercise like this: $\displaystyle \lim_{x \to 0}\frac{\ln (1 + x^2)}{x } = x$

How does this all fit together so I actually get to x? I need this explained in baby steps, it just won't ring a bell at the moment