Problem Set 3 — Limits and Continuity

Purpose

This problem set develops your understanding of variations on the basic idea of a limit, for example one-sided limits or infinite limits. It also explores the notion of continuity.

This problem set addresses the following learning outcomes:

• Compute limits and derivatives of algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and piecewise defined functions (outcome 1)
• Determine the continuity and differentiability of a function at a point and on a set (outcome 3).

Background

This problem set uses ideas from sections 2.2 through 2.4 of our textbook. We discussed this material in classes between September 14 and 18.

Activity

Solve the following problems:

Question 1

Find

and

Does \(\lim_{x \to -2} \frac{x^2-4}{|x+2|}\) exist? If so, what is it? If not, why not?

Question 2

Consider the function \(f(x)\) defined piecewise by

Part A

Find \(\lim_{x \to 0^{-}} f(x)\).

Part B

Find \(\lim_{x \to 0^{+}} f(x)\).

Part C

Does \(\lim_{x \to 0} f(x)\) exist? If so, what is its value? If not, why not?

Part D

Is \(f(x)\) continuous at \(x = 0\). Explain why or why not.

Question 3

(Based on exercise 154 in section 2.4 of our textbook.)

Using the graph in exercise 154 (at the top of page 192 in our PDF of the book), say whether the function \(f(x)\) is or is not continuous at each of the following \(x\) values:

1. \(x = -2\)
2. \(x = -1\)
3. \(x = 0\)
4. \(x = 1\)
5. \(x = 2\)

For each value at which the function is discontinuous, say what part of the definition of continuity is violated, and identify the discontinuity as a jump, infinite, or removable discontinuity.

Question 4

Newton’s law of gravitation implies that the acceleration due to gravity at the surface of a star of mass \(m\) (measured in kilograms) is

where \(r\) is the radius of the star in meters and \(G\) is a constant approximately equal to \(6.67 \times 10^{-11}\ \mathrm{N m^2 / kg^2}\).

Imagine that a spaceship (evidently a very hardy one) is skimming the surface of a star that starts collapsing. The star collapses in such a way that its mass remains constant, but gets packed into an ever smaller radius, and throughout the collapse the spaceship keeps flying right along the surface of the star. Calculate the acceleration due to the star’s gravity on the spaceship in the limit as the star’s radius approaches 0.

To what extent do you think you can rely on this mathematical result as a model of physical reality? (For brownie points, but not actual grade points, what is a star called if it collapses to the point that its radius approaches 0?)

Follow-Up

I will grade this exercise during one of your weekly individual meetings with me. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.