Stewart 9th - 1.1, 1.6 (8Q Hard)

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side \(x\) at each corner and then folding up the sides as in the figure. Express the volume \(V\) of the box as a function of \(x\).

\( \operatorname*{lim}\limits_{t \rightarrow 0} \left(\dfrac{1}{t \sqrt{1 + t}} - \dfrac{1}{t}\right) \)

The figure shows a fixed circle \(C_1\) with equation \((x - 1)^2 + y^2 = 1\) and a shrinking circle \(C_2\) with radius \(r\) and center the origin. \(P\) is the point \((0, r)\), \(Q\) is the upper point of intersection of the two circles, and \(R\) is the point of intersection of the line \(P Q\) and the \(x\)-axis. What happens to \(R\) as \(C_2\) shrinks, that is, as \(r \rightarrow 0^+\)?

If \(f(x) = \begin{cases} x^2 \text{if x is rational} \\ 0 \text{if x is irrational} \end{cases}\), prove that \(\operatorname*{lim}\limits_{x \rightarrow 0} f(x) = 0\).

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area \(A\) of the window as a function of the width \(x\) of the window.

\( \operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{\dfrac{1}{(x + h)^2} - \dfrac{1}{x^2}}{h} \)

Evaluate \(\operatorname*{lim}\limits_{x \rightarrow 2} \dfrac{\sqrt{6 - x} - 2}{\sqrt{3 - x} - 1}\).

Is there a number \(a\) such that \(\operatorname*{lim}\limits_{x \rightarrow -2} \dfrac{3 x^2 + a x + a + 3}{x^2 + x - 2}\) exists? If so, find the value of \(a\) and the value of the limit.