AP Calculus AB: Solving Limits Question on Continuity

[Chirag B]

I tutor an AP Calculus AB class at the library every week, and I recently gave the students a test on Limits and Continuity. The test was fairly straightforward, but some of the free-response questions were somewhat difficult. The students seemed to get the answers fine, but my friend/mentor seemed to disagree with the answer to one of the questions.

The problem in question is the following limits question: lim_{x→0} \frac{\sqrt{ax + b} – 2}{x} = 1

Given this equality, students were asked to find the values of a and b.

My solutions were a = 4 and b = 4. However, this was disputed by my friend/mentor, who claimed that if x ≠ 0.

Can somebody settle this dispute? What are the values of a and b?

 

[mathman]

In order for the function to have a limit, the numerator must -> 0 as x -> 0. This forces √b = 2 (or b = 4). Using L'Hopital's rule, a = 2√b = 4.