91Ó°ÊÓ

Use the precise definition of a limit to prove the following limits. $$\lim _{x \rightarrow 4} \frac{x^{2}-16}{x-4}=8 \text { (Hint: Factor and simplify.) }$$

Evaluating limits analytically Evaluate the following limits or state that they do not exist. a. \(\lim _{x \rightarrow 3^{+}} \frac{(x-1)(x-2)}{(x-3)}\) b. \(\lim _{x \rightarrow 3^{-}} \frac{(x-1)(x-2)}{(x-3)}\) c. \(\lim _{x \rightarrow 3} \frac{(x-1)(x-2)}{(x-3)}\)

For the following position functions, make a table of average velocities similar to those in Exercises \(19-20\) and make a conjecture about the instantaneous velocity at the indicated time. $$s(t)=-16 t^{2}+80 t+60 \quad \text { at } t=3$$

Use the graph of \(f\) in the figure to find the following values, if they exist. If a limit does not exist, explain why. a. \(f(1)\) b. \(\lim _{x \rightarrow 1^{-}} f(x)\) c. \(\lim _{x \rightarrow 1^{+}} f(x)\) d. \(\lim _{x \rightarrow 1} f(x)\)

Applying limit laws Assume \(\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3\) and \(\lim _{x \rightarrow 1} h(x)=2 .\) Compute the following limits and state the limit laws used to justify your computations. $$\lim _{x \rightarrow 1}\left[\frac{f(x) g(x)}{h(x)}\right]$$

Determine the following limits. $$\lim _{x \rightarrow-\infty}\left(-3 x^{16}+2\right)$$

Use Theorem 10 to determine the intervals on which the following functions are continuous. $$p(x)=4 x^{5}-3 x^{2}+1$$

For the following position functions, make a table of average velocities similar to those in Exercises \(19-20\) and make a conjecture about the instantaneous velocity at the indicated time. $$s(t)=20 \cos t \quad \text { at } t=\pi / 2$$

Determine the following limits. $$\lim _{x \rightarrow-\infty} 2 x^{-8}$$

Use the precise definition of a limit to prove the following limits. $$\lim _{x \rightarrow 3} \frac{x^{2}-7 x+12}{x-3}=-1$$