Evaluate \( \lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}-\cot ^{2} x\right) \)

Determine el área encerrada entre las curvas \( y = x^2 \) y \( y = 4 - x^2 \), dado que se intersectan en más de dos puntos.

Calculate the volume of the solid formed by rotating the area between \( y = \sqrt{x} \) and \( y = 0 \) from \( x = 0 \) to \( x = 1 \) about the line \( y = -1 \) utilizing the washer method.

d) \( \lim _{x \rightarrow 1} \frac{\ln x}{x^{2}-1}= \)

Compute the average value of \( f(t) = \sin(t) + \cos(t) \) over the interval \([0, \pi]\).

\( \lim _ { x \rightarrow 9 } ^ { 9 : 03 } ( \frac { \sqrt { x - 9 } } { \sqrt { x } - 3 } ) \)