Question

I have a few more calculus questions attached in the post.

Answer 1

If you post one problem at a time people may be more willing to help.

For the first problem, rationalize the numerator.

For the second problem, split it as a product of two limits: sin(x)/x * sin(x).

Answer 2

For the first problem, multiply numerator and denominator by:
\(\sqrt{x+\Delta x - 9} + \sqrt{x - 9} \)

Answer 3

\[
\frac{\sqrt{x+\Delta x - 9} - \sqrt{x - 9}}{\Delta x} * \frac{\sqrt{x+\Delta x - 9} + \sqrt{x - 9}} { \sqrt{x+\Delta x - 9} + \sqrt{x - 9}} =
\frac{x+\Delta x - 9 - x + 9}{{ \Delta x(\sqrt{x+\Delta x - 9} + \sqrt{x - 9}})} = \\
\frac{1}{{ \sqrt{x+\Delta x - 9} + \sqrt{x - 9}}}
\]
Now it is easy to take the limit.