Calculus I :: Lab 03

Enter in your name, I need to know who to give the grades to.

Let \(\displaystyle f(x) = \frac{\sin(kx)}{x}\), where \(k\) is some constant.

1. Graph the function \(f(x)\), using a slider for \(k\).
2. Make a conjecture about \(\displaystyle \lim_{x\to 0} f(x)\).

Repeat Problem 1 for \(\displaystyle g(x) = \frac{\cos(kx)}{x}\).

Let \(\displaystyle h(x) = (x-3)^2\cos\left(\frac{7}{x-3}\right)\), and consider \(\lim_{x\to 3} h(x)\).

1. Find functions \(u(x)\) and \(l(x)\), and an interval \((a,b)\) so that the Squeeze Theorem can be applied. That is show (graphically) that \(l(x) \leq f(x) \leq u(x)\) for all \(x \in (a,b)\), except possibly \(x=3\).
2. Apply the Squeeze Theorem to find the limit.

Repeat Problem 3 for \(\displaystyle j(x) = (x-\pi/2) \sin(\sec x)\) and \(\lim_{x\to \pi/2} j(x)\).

Discuss \(\displaystyle \lim_{x \to \infty} \frac{\sin(2\pi x)}{e^x}\). Can you apply the Squeeze Theorem? Why or why not?