91Ó°ÊÓ

Prove the statement using the \(\varepsilon, \delta\) definition of a limit. $$\lim _{x \rightarrow a} c=c$$

\(37-38=\) Find \((a) f+g,(\) b) \(f-g,(\) c) \(f g,\) and \((d) f / g\) and state their domains. $$f(x)=\sqrt{3-x}, \quad g(x)=\sqrt{x^{2}-1}$$

Suppose \(f\) is continuous on \([1,5]\) and the only solutions of the equation \(f(x)=6\) are \(x=1\) and \(x=4 .\) If \(f(2)=8\) explain why \(f(3)>6\)

Estimate the horizontal asymptote of the function $$f(x)=\frac{3 x^{3}+500 x^{2}}{x^{3}+500 x^{2}+100 x+2000}$$ by graphing \(f\) for \(-10 \leqslant x \leqslant 10 .\) Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?

\(39-42\) . Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. $$x^{4}+x-3=0, \quad(1,2)$$

\(39-44=\) Find the functions (a) \(f \circ g,(\mathrm{b}) g \circ f,(\mathrm{c}) f \circ f,\) and (d) \(g \circ g\) and their domains. $$f(x)=x^{2}-1, \quad g(x)=2 x+1$$

Prove the statement using the \(\varepsilon, \delta\) definition of a limit. $$\lim _{x \rightarrow 0} x^{2}=0$$

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 0-5} \frac{2 x-1}{\left|2 x^{3}-x^{2}\right|}$$

Find the domain and sketch the graph of the function. $$f(x)=\left\\{\begin{array}{ll}{x+2} & {\text { if } x<0} \\ {1-x} & {\text { if } x \geqq 0}\end{array}\right.$$

\(39-42\) . Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. $$\sqrt[3]{x}=1-x, \quad(0,1)$$