Estimate the instantaneous rate of change of T with respect to time at 8 P.M. by measuring the slope of a tangent.

In the following exercise find the limit.

$$\lim _{x \rightarrow^ \infty}\left(\sqrt{9 x^2+x}-3 x\right)$$

In the following exercise find the limit.

$$\lim_{x \rightarrow^-\infty}\left(x+\sqrt{x^2+2 x}\right)$$

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$$f(x)=1-3 x^2$$

Explain, why the function is continuous at every number in its domain. State the domain. $R(x)=x^{2}+\sqrt{2 x-1}$

Determine the infinite limit.

$$\lim _{x \rightarrow 5^{-}} \frac{6}{x-5}$$

Explain, using the mentioned theorems, why the function $h(x)=\sqrt[5]{x-1}\left(x^2-2\right)$ is continuous at every number in its domain. State the domain.

Prove each statement using the $\varepsilon, \delta$ definition of limit.

$$\lim _{x \rightarrow a} x=a$$

Use graphs to discover the asymptotes of the curve. Then prove what you have discovered.

$$y=\dfrac{\cos ^2 x}{x^2}$$

Determine the infinite limit.

$$\lim _{x \rightarrow 3} \dfrac{1}{(x-3)^8}$$

Evaluate the limit, if it exists. $\lim _{x \rightarrow 7} \frac{\sqrt{x+2}-3}{x-7}$

Each limit represents the derivative of some function f at some number a. State f and a in each case.

$$\lim _{t \rightarrow 0} \dfrac{\sin \left(\dfrac{\pi}{2}+t\right)-1}{t}$$

Use graphs to discover the asymptotes of the curve. Then prove what you have discovered.

$y=\frac{\cos ^2 x}{x^2}$

Each limit represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in each case.

$$\displaystyle\lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h}$$