91Ó°ÊÓ

A function is continuous from the right at \(x=a\) if \(\lim _{x \rightarrow a^{+}} f(x)=f(a) .\)Determine whether \(f(x)\) is continuous from the right at \(x=2.\) $$f(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } x<2 \\ 3 x-2 & \text { if } x>2 \end{array}\right.$$

Use graphical and numerical evidence to conjecture the value of the limit. Then, verify your conjecture by finding the limit exactly. $$\lim _{x \rightarrow \infty}(\sqrt{x^{2}+3}-x)$$

Use the given position function \(f(t)\) to find the velocity at time \(t=a\). $$f(t)=t^{2}+2, a=0$$

Use the given position function \(f(t)\) to find the velocity at time \(t=a\). $$f(t)=t^{3}, a=0$$

A metal washer of (outer) radius \(r\) inches weighs \(2 r^{2}\) ounces. A company manufactures 2 -inch washers for different customers who have different error tolerances. If the customer demands a washer of weight \(8 \pm \varepsilon\) ounces, what is the error tolerance for the radius? That is, find \(\delta\) such that a radius of \(r\) within the interval \((2-\delta, 2+\delta)\) guarantees a weight within \((8-\varepsilon\) \(8+\varepsilon)\)

Use graphical and numerical evidence to conjecture the value of the limit. Then, verify your conjecture by finding the limit exactly. $$\lim _{x \rightarrow \infty}(\sqrt{5 x^{2}+4 x+7}-\sqrt{5 x^{2}+x+3})$$

Use the given position function \(f(t)\) to find the velocity at time \(t=a\). $$f(t)=t^{3}, a=1$$

A fiberglass company ships its glass as spherical marbles. If the volume of each marble must be within \(\varepsilon\) of \(\pi / 6,\) how close does the radius need to be to \(1 / 2 ?\)

Use graphical and numerical evidence to conjecture the value of the limit. Then, verify your conjecture by finding the limit exactly. $$\lim _{x \rightarrow-\infty}\left(1+\frac{3}{x}\right)^{2 x}$$

Suppose that \(f(x)=\frac{g(x)}{h(x)}\) and \(h(a)=0 .\) Determine whether each of the following statements is always true, always false, or maybe true/maybe false. Explain. (a) \(\lim _{x \rightarrow a} f(x)\) does not exist. (b) \(f(x)\) is discontinuous at \(x=a.\)