7.) We see that and so the limit is of the indeterminate form 0/0, so applying L'Hôpital's rule, we may state the limit as:

Now, use the slopes of the given tangent lines to complete the computation.

Your reworking looks good, you were probably penalized the first time for using the equations of the tangent lines in the limit.

9.) Differentiating with respect to , we find the velocity function:

The particle is at rest when .

This implies:

where

So let , and we must then have