Evaluate the following limits: 1. lim x->1 (x-1)/(x-1) 2. lim x->1 (x2-1)/(x2-1) 3. lim x->3 (x3-27)/(x-9) 4. lim x->0 ((3+x)2-9)/x

This is an indeterminate form.

For $x \neq 1$, the expression $\frac{x-1}{x-1}$ simplifies to 1.

The limit is $\boxed{1}$.

2.

This is an indeterminate form.

For $x \neq 1$, the expression $\frac{x^2-1}{x^2-1}$ simplifies to 1.

The limit is $\boxed{1}$.

3.

This is not an indeterminate form.

The limit is $\boxed{0}$.

4.

This is an indeterminate form.

The limit is $\boxed{6}$.

5.

This is an indeterminate form.

The numerator is a difference of cubes: $x^3-1 = (x-1)(x^2+x+1)$.
The denominator is a quadratic: $x^2-3x+2 = (x-1)(x-2)$.

The limit is $\boxed{-3}$.

6.

This is an indeterminate form.

The numerator is a quadratic: $x^2-7x+10 = (x-2)(x-5)$.
The denominator is a difference of squares: $x^2-25 = (x-5)(x+5)$.

The limit is $\boxed{\frac{3}{10}}$.

For questions 7-11, we will find the derivative $\frac{dy}{dx}$.

7.

The derivative is $\boxed{\sec x \tan x}$.

8.

The derivative is $\boxed{-\csc x \cot x}$.

9.

Let $u = x^2+4$. Then $y = u^4$.

The derivative is $\boxed{8x(x^2+4)^3}$.

10.

The derivative is $\boxed{\cos x}$.

11.

Let $u = x^2+4$. Then $y = u^3$.

The derivative is $\boxed{6x(x^2+4)^2}$.