(f) \( \lim _{x \rightarrow \frac{\pi}{4}} \frac{\sin x-\cos x}{1-\tan x} \) (g) \( \lim _{x \rightarrow 0} \frac{x-\sin 2 x}{x+\sin 3 x} \) (h) \( \lim _{x \rightarrow 0} \frac{1-\sqrt{\cos x}}{x^{2}} \) (i) \( \lim _{x \rightarrow \infty}\left(\frac{x-1}{x+3}\right)^{x+2} \)

\( e = \frac { 2 } { \theta + 1 } . \quad \frac { d e } { d \theta } = - \frac { 2 } { ( \theta + 1 ) ^ { 2 } } \)

4 A particle \( P \) is attached to one end of a light elastic spring. The other end of the spring is attached to a fixed point \( A \) and \( P \) hangs freely in equilibrium vertically below \( A \). At time \( t=0, P \) is projected vertically downwards with speed \( u \). A resistance proportional to the speed of \( P \) acts on \( P \). The motion of \( P \) can be modelled using the differential equation \( \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+4 k \frac{\mathrm{~d} x}{\mathrm{~d} t}+4 k^{2} x=0 \) where \( x \) is the displacement of \( P \) from its equilibrium position at time \( t \) and \( k \) is a positive constant. a Find an expression for \( x \) in terms of \( u, t \) and \( k \). (6) Find the time at which \( P \) comes to instantaneous rest.

For \( x \quad>\quad 0 \), which answer best describes the curve of the function \( f(x)=x^{\frac{1}{2}} ? \) Positive values of \( f(x) \); steep near 0 , but flattening as \( x \) approaches \( \infty \) Negative values of \( f(x) \); steep near 0, but flattening as \( x \) approaches \( \infty \) Positive values of \( f(x) \); flat near 0, but steepening as \( x \) approaches \( \infty \) Negative values of \( f(x) \); flat near 0, but steepening as \( x \) approaches \( \infty \) None of the above.

(2) \( \int \frac{3}{2} a^{-x} d x \)

2. Es la opción que contiene la derivada de la función que representa al volumen de la caja sin tapa. \( V^{\prime}(L)=-12 L^{3}-120 L^{2}+200 \) \( V^{\prime}(L)=12 L^{2}-120 L+200 \) \( V^{\prime}(L)=12 L^{2}+120 L-200 \) Opción 1 Opción 2