Hello Calculus students!
For this (partial) week, you should be re-reading sections 1.1-1.2 and reading sections 1.3 and (at least parts of) 1.4.
In addition to working on your 5 question assignment via WeBWorK (Assignment 2 – The Clone Wars), this Friday you will also need to turn in a handwritten assignment. This will consist of the following problem (that was first described in class):
Handwritten Assignment 1
1(a) Consider the function . Use one-sided limits to conclude that the instantaneous velocity of does not exist at .
1(b) Draw a graph of the function and explain (in a sentence or two) how the graph corroborates your work in part (a).
1(c) Explain why the function is continuous at and, in fact, why it is continuous at every value.
I also want to use this blog post to include two more examples of (algebraically) computing some limits.
Example (1). Let’s consider the function
This function has as its domain all values where and . If we want to compute the limit we cannot just plug in since this produces an expression with zero in the denominator. Instead, we must first algebraically manipulate the formula. Taking a hint from today’s class we can multiply the top and bottom of the first fraction by . In fact, we can also factor the denominator for the second expression:
Now that this expression has a common denominator, we can write
We should be careful and note that the cancellation used above works provided . Since we are now taking a limit as , we are allowed to do exactly this kind of simplification. We find