Basically, you keep taking the derivative of the numerator and the denominator until you can plug in the value for the limit.

For the first one, for example:

limit of ((e^x)-1)/sin(7x) as x --> 0

take derivatives of (e^x)-1 (which is e^x) and sin(7x) (which is (7cos(7x))

so now it looks like this:

limit of (e^x)/7cos(7x) as x --> 0

you can now plug in 0 without division by 0

and you get

1/7

I hope this helps with the rest