The problem involves evaluating the limit of a function as x approaches zero, specifically the expression ##\lim_{x->0}\frac{\sqrt{ax+b}-5}{x}##, and determining the values of a and b such that this limit equals ##\frac{1}{2}##. The context is rooted in calculus, particularly in the application of limits and potentially L'Hôpital's rule.
The discussion includes various approaches to the problem, with some participants suggesting specific values for b and a based on their reasoning. There is an indication that productive lines of inquiry are being explored, particularly around the implications of the limit and the relationships between a and b.
Participants are working under the constraints of the limit approaching zero and the requirement for the limit expression to yield a specific value. The original poster expresses difficulty in progressing after an initial application of L'Hôpital's rule, indicating potential gaps in understanding the necessary conditions for the limit.
What must be true for ##\displaystyle \ \frac{\sqrt{ax+b}-5}{x}\ ## if L'Hôpital's rule can be applied? In particular what must be true of ##\displaystyle \ \lim_{x->0}(\sqrt{ax+b}-5) \ ?##terryds said:Homework Statement
If a and b satisfy ##\lim_{x->0}\frac{\sqrt{ax+b}-5}{x} = \frac{1}{2}##, then a+b equals...
A. -15
B. -5
C. 5
D. 15
E. 30
Homework Equations
L'hospitalThe Attempt at a Solution
By using L'hospital, I get b=a^2
Then, I got stuck.. Substituting b=a^2 into the limit equation, but I still can't cancel out the x which is the cause of zero denominator..
Please help
SammyS said:What must be true for ##\displaystyle \ \frac{\sqrt{ax+b}-5}{x}\ ## if L'Hôpital's rule can be applied? In particular what must be true of ##\displaystyle \ \lim_{x->0}(\sqrt{ax+b}-5) \ ?##