91Ó°ÊÓ

use the Intermediate Value Theorem to verify that \(f(x)\) has a zero in the given interval. Then use the method of bisections to find an interval of length \(1 / 32\) that contains the Zero. $$f(x)=\cos x-x,[0,1]$$

Prove that the limit is correct using the appropriate definition (assume that \(k\) is an integer). $$\lim _{x \rightarrow 5} \frac{4}{(x-5)^{2}}=\infty$$

Use graphical and numerical evidence to conjecture a value for the indicated limit. $$\lim _{x \rightarrow-1} \frac{x-\cos (\pi x)}{x+1}$$

A parking lot charges \(\$ 2\) for each hour or portion of an hour, with a maximum charge of \(\$ 12\) for all day. If \(f(t)\) equals the total parking bill for \(t\) hours, sketch a graph of \(y=f(t)\) for \(0 \leq t \leq 24 .\) Determine the limits \(\lim _{t \rightarrow 3.5} f(t)\) and \(\lim _{t \rightarrow 4} f(t),\) if they exist.

use the Intermediate Value Theorem to verify that \(f(x)\) has a zero in the given interval. Then use the method of bisections to find an interval of length \(1 / 32\) that contains the Zero. $$f(x)=e^{x}+x,[-1,0]$$

Explain how to determine \(\lim _{x \rightarrow a} f(x)\) if \(g\) and \(h\) are polynomials and \(f(x)=\left\\{\begin{array}{ll}g(x) & \text { if } x < a \\ c & \text { if } x=a \\ h(x) & \text { if } x > a\end{array}\right.\)

Prove that the limit is correct using the appropriate definition (assume that \(k\) is an integer). $$\lim _{x \rightarrow-4} \frac{-6}{(x+4)^{6}}=-\infty$$

Use graphical and numerical evidence to conjecture a value for the indicated limit. $$\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}$$

Identify a specific \(\varepsilon>0\) for which no \(\delta>0\) exists to satisfy the definition to limit. $$f(x)=\left\\{\begin{array}{ll} 2 x & \text { if } x<1, \lim _{x \rightarrow 1} f(x) \neq 2 \\ x^{2}+3 & \text { if } x>1 \end{array}\right.$$

Evaluate each limit and justify each step by citing the appropriate theorem or equation. (b) \(\lim _{x \rightarrow 0} \frac{x-2}{x^{2}+1}\) (a) \(\lim _{x \rightarrow 2}\left(x^{2}-3 x+1\right)\)