Chapter 3

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow-5}\left(4+2 x^{2}\right) $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 0} \frac{e^{-x}-e^{x}}{e^{-x}+1} $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 2}\left(8 x^{3}-2 x+4\right) $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow-2} \frac{1}{\sqrt{5 x^{2}-4}} $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 3}\left(2 x^{2}-\frac{1}{x}\right) $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 1} \frac{1}{\sqrt{3-2 x^{2}}} $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow+2}\left(\frac{x^{2}}{2}-\frac{2}{x^{2}}\right) $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+9}-3}{x^{2}} $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow-3} \frac{x^{3}-20}{x+1} $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 0} \frac{5-\sqrt{25+x^{2}}}{2 x^{2}} $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 1} \frac{x^{3}+1}{x+2} $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 0} \ln (1-x) $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 3} \frac{3 x^{2}+1}{2 x-3} $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 1} \ln \left[e^{x} \cos (x-1)\right] $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow-2} \frac{1+x}{1-x} $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 1} \frac{1-x^{2}}{1-x} $$

Fungal Growth As a fungus grows, its rate of growth changes. Young fungi grow exponentially, while in larger fungi growth slows, and the total dimensions of the fungus increase as a linear function of time. You want to build a mathematical model that describes the two phases of growth. Specifically if \(R(t)\) is the rate of growth given as a function of time, \(t\), then you model $$ R(t)=\left\\{\begin{array}{ll} 2 e^{t} & \text { if } 0 \leq t \leq t_{c} \\ a & \text { if } t>t_{c} \end{array}\right. $$ where \(t_{c}\) is the time at which the fungus switches from exponential to linear growth and \(a\) is a constant. (a) For what value of \(a\) is the function \(R(t)\) continuous at \(t=t_{c}\) ? (Your answer will include the unknown constant \(t_{c}\) ). (b) Assume that \(t_{c}=2 .\) Draw the graph of \(R(t)\) as a function of \(t\)

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{u \rightarrow 2} \frac{4-u^{2}}{2-u} $$

Panting in Animals Animals use different strategies to control their internal temperature depending on how hot they are. When the core temperature of a dog, duck, or cat exceeds a critical value, it will start to pant (make quick, gasping breaths that increase evaporation of water from the tongue and mouth). Vieth (1989) studied heat loss as a function of the ducks' core temperature, \(T\). She found that different functions described heat loss below the temperature at which the ducks started to pant and above this temperature. If \(H(T)\) is the rate of heat loss: $$ H(T)=\left\\{\begin{array}{ll} 0.6 & \text { if } T \leq T_{c} \\ 4.3 T-183 & \text { if } T>T_{c} \end{array}\right. $$ (here \(T\) is measured in \({ }^{\circ} \mathrm{C}\) and \(H(T)\) in watts per \(\mathrm{kg}\) of body mass \()\) (a) Calculate the value of \(T_{c}\) that makes \(H(T)\) continuous for all \(T\). (b) Draw the graph of the function \(H(T)\) over the normal body temperature range for ducks: \(41^{\circ} \mathrm{C} \leq T \leq 44^{\circ} \mathrm{C}\).

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 3} \frac{x^{2}-2 x-3}{x-3} $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 1} \frac{(x-1)^{2}}{x^{2}-1} $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 2} \frac{2-x}{x^{2}-4} $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow-4} \frac{x+4}{16-x^{2}} $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 1} \frac{1-2 x+x^{2}}{1-x} $$