Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). \(\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{(2 + h)^5 - 32}{h}\), \(h = \pm 0.5, \pm 0.1, \pm 0.01, \pm 0.001, \pm 0.0001\)

Consider the function \(f(x) = \tan\left(\dfrac{1}{x}\right)\).

Explain in your own words what is meant by the equation \(\operatorname*{lim}\limits_{x \rightarrow 2} f(x) = 5\). Is it possible for this statement to be true and yet \(f(2) = 3\)? Explain.

Use the Intermediate Value Theorem to show that there is a solution of the equation in the given interval. \(\dfrac{2}{x} = x - \sqrt{x}\), \((2, 3)\)

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(B(u) = \sqrt{3u - 2} + \sqrt{2u - 3}\)

Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x) = \begin{cases} \cos x & \quad \text{if } x < 0 \\ 0 & \quad \text{if } x = 0 \\ 1 - x^2 & \quad \text{if } x > 0 \end{cases}\), \(a = 0\)

If \(f(x) = x^2 + 10 \sin x\), show that there is a number \(c\) such that \(f(c) = 1000\).

Suppose \(f\) and \(g\) are continuous functions such that \(g(2) = 6\) and \(\operatorname*{lim}\limits_{x \rightarrow 2} [3 f(x) + f(x) g(x)] = 36\). Find \(f(2)\).

Find the given derivative by finding the first few derivatives and observing the pattern that occurs. \(D^{103} \cos 2x\)

Write the composite function in the form \(f(g(x))\). [Identify the inner function \(u = g(x)\) and the outer function \(y = f(u)\).] Then find the derivative \(\dfrac{d y}{d x}\). \(y = \tan(x^2)\)

\( A(t) = \dfrac{1}{(\cos t + \tan t)^2} \)

\( G(z) = (1 - 4z)^2 \sqrt{z^2 + 1} \)

The function \(f(x) = \sin(x + \sin 2x)\), \(0 \leq x \leq \pi\), arises in applications to frequency modulation (FM) synthesis.

\(y \sin 2x = x \cos 2y\), \(\quad \left(\dfrac{\pi}{2}, \dfrac{\pi}{4}\right)\)

If \(f(x) + x^2 [f(x)]^3 = 10\) and \(f(1) = 2\), find \(f'(1)\).

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. \(x^2 + y^2 = r^2\), \(\quad a x + b y = 0\)

Use implicit differentiation to find \(\dfrac{d y}{d x}\) for the equation \(\dfrac{x}{y} = y^2 + 1\) (where \(y \neq 0\)). Then show that you get the same answer when you perform implicit differentiation on the equivalent equation \(x = y^3 + y\) (where \(y \neq 0\)).

\(y^2(6 - x) = x^3\), \(\quad (2, \sqrt{2})\) (cissoid of Diocles)