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) \ ?##
A limit problem is a mathematical problem that involves finding the value that a function approaches as its input approaches a certain value. In other words, it is a way to analyze the behavior of a function near a specific point.
To solve a limit problem, you must first determine the type of limit (left, right, or two-sided) and then apply the appropriate limit rules. If the function is continuous at the given point, you can simply plug in the value and evaluate. If the function is not continuous, you may need to use algebraic techniques such as factoring or rationalizing to simplify the expression before evaluating the limit.
The purpose of finding a limit is to better understand the behavior of a function. Limits can help determine if a function is continuous at a certain point, identify vertical or horizontal asymptotes, and evaluate indeterminate forms. They are also used in calculus to calculate derivatives and integrals.
If the limit problem includes a variable in the expression, you can use algebraic techniques to simplify the expression before evaluating the limit. This may involve factoring, rationalizing, or using trigonometric identities. Once the expression is simplified, you can then plug in the given value to evaluate the limit.
Solving limit problems can help in various real-world applications, such as determining maximum or minimum values, calculating rates of change, and analyzing the behavior of a system. For example, in physics, limits can be used to calculate velocity and acceleration, and in economics, they can be used to analyze supply and demand functions.