Question 1 introduces some terms I don't discuss in my book (and which you are unlikely to see again). I think you can pretty much guess which graph represents which discontinuity.
A removable discontinuity is one where the graph actually converges on one specific place (i.e. the limit of the function exists as x approaches a), but there is a hole in the graph (f(a) does not exist). Thus, you could remove the discontinuity if you could only plot a point to fill in that hole.
A jump discontinuity is where the graph jumps from one location to another (a break in the graph). i.e. The limit as x approaches a does not exist because the left limit does not lead to the same place as the right limit. It is irrelevant whether f(a) exists or not.
An infinite discontinuity is where the graph flies off to infinity or negative infinity causing a discontinuity. It will also cause a break in the graph much like a discontinuity, but the difference is the left or right limits are infinite rather than finite.
Question 2 is more of the same. This time you will need to actually compute the limits as x approaches 3 to see what is actually happening. The limits will help you visualize what the graph looks like as x approaches 3.
Question 3 is using the Intermediate Value Theorem as I teach in Lesson 3 and illustrate in questions 4 and 5.
Question 4 is classic continuity stuff, similar to my question 3 in Lesson 3.
Question 5 is some challenging infinity limits. Be sure to have studied that section of Lesson 2 very closely. Especially be aware of how to deal with the square root of x-squared. Look at questions 10 to 14 in my Lecture Problems and questions 57 to 74 in my Practise Problems.
Question 6 is like my question 15 in Lesson 2. You must figure out where the function is k/0 for vertical asymptotes and compute BOTH the limits as x approaches infinity and negative infinity for horizontal asymptotes. Note that each infinity limit is looking for its own horizontal asymptote. You could end up with no horizontal asymptotes, one, or two.
Question 7 does not specifically tell you to use the definition of derivative, so you can actually use differentiation rules to find the derivative and proceed to get the equation of the tangent. However, because they mention sections 2.7 and 2.8, that implies they want you to use definition of derivative in this question (as I teach in Lesson4 4).
Question 8 is clearly a definition of derivative question.
Question 9 is a runthrough of all the differentiation rules taught in Lesson 5 of my book.
Question 10 is a trig limit as taught in Lesson 2 of my book, question 16 and Practise Problems 75, 76, and 77 .
