Evaluate the  $\lim_{x\to 4}f\left(x\right)$,  where\begin{align*} {f\left(x\right)}&={\begin{cases} {5\cos x}&{x\leq3.9}\\ {3x-4}&{x>3.9}\\ \end{cases}\,} \end{align*}

Evaluate   \begin{gather*} {\lim_{x\to 4}\frac{3x^{2}-12x}{x^{2}-16}} \end{gather*}

Evaluate   \begin{gather*} {\lim_{x\to 1}\frac{{\sqrt[\,]{x+4}}-{\sqrt[\,]{6-x}}}{x-1}\,} \end{gather*}

Let   $$f\left(x\right)=\begin{cases} {2x^{4}-7}&{x<0}\\ {0}&{x=0}\\ {\sin x+2}&{x>0}\\ \end{cases}$$Evaluate the limit   $\lim_{x\to 0^{-}}f\left(x\right)$.

Evaluate
\begin{gather*} {\lim_{x\to -\infty}\frac{1-2x-4x^{2}}{4x^{2}+5}} \end{gather*}

Find all values of  $c$  for which the following function is continuous.
$$f\left(x\right)=\begin{cases} {x^{2}-10}&{x\leq c}\\ {2-x}&{x>c}\\ \end{cases}$$

Use the definition of the derivative to evaluate the derivative of the function$$f\left(x\right)=\frac{3}{x+1}$$

Differentiate  $$\dfrac{2-x^{2}}{3x^{2}+7}$$

Differentiate  $$\left(e^{x}-1\right)\left(2e^{x}+1\right)$$

Using the derivative, determine whether   $y\left(x\right)=e^{3x}-5$  describes a straight line.

Evaluate 
\begin{gather*} {\lim_{x\to 0}\frac{\sin x^{37}+3x^{7}e^{x^{99}}}{\sin^{7}x}} \end{gather*}

Position of a particle at time  $t$  is given by  $s\left(t\right)=e^{t^{3}\VPlusc t^{2}\VPlusd t}$. For which values of  $t$  is the velocity of the particle zero?

Compute the derivative of   $$\cos\left(x^{2}+{\sqrt[]{x^{2}+1}}\right)$$

Using the identity  $\sin2x=2\sin x\cos x$, derive a similar identity for  $\cos2x$.

Find the derivative of   $$\dfrac{\ln x}{x^{6}}$$

Find the derivative of
$$y=\left(1-6x\right)^{\cos x}\,$$

Find  $\diff{y}{x}$  if   $xy+e^{x}+e^{y}=1$.

Differentiate  $$\sec^{-1}\left(-x^{2}-2\right)$$

Find an approximation to  ${\sqrt[\,]{8.98}}$  using the linearization of the function  $f\left(x\right)={\sqrt[\,]{x+5}}$ .

Find $\diff{z}{t}$ at $(x,y)=(4,2)$ if $z=\frac{x^{2}}{2}+\frac{y^{2}}{8}$, $\diff{x}{t}=-2$, and $\diff{y}{t}={6}$.

A water tank of the shape of an inverted circular cone with base radius $2\um$ and height $6\um$ is pumped water into at a rate $3\,\um^{3}/\umin$. At what rate the water is rising when the water is $4\um$ deep?
Let  $V$  be the volume of the water,  $r$,  and  $h$  be the radius of the surface, and the height at time  $t$,  where $t$ is measured in minutes.

A cyclist A is approaching an intersection from west at $50\ukm/\uhr$. Another cyclist B is approaching the same intersection along a perpendicular direction from south at $60\ukm/\uhr$.

At what rate are the cyclists approaching each other when the first cyclist is  $x=0.3\ukm$ and the other is  $y=0.4\ukm$ from the intersection, respectively?

Find the area of the largest rectangle inscribed in a semicircle of radius $r$.