Rationalize the denominator of ##\displaystyle \ \frac{f(x)−f(2)}{\sqrt{x\,}-\sqrt{2\,}}\ .##i_m_mimi said:Homework Statement
Suppose that f' (2) = 3. Find the limit as x approaches 2 of [f(x)−f(2)]/[sqrt(x) - sqrt(2)]
Answer: 6*sqrt 2
Homework Equations
The Attempt at a Solution
f'(x) = lim h->0 = [f(a +h) - f(a)]/h = slope [f(x)-f(2)]/ x-2
a = 2
i would think that the limit = 3? wrong answer
tried graphing it, but no original function is given
sqrtx and sqrt2 confuses me and not sure how to do it.
thank you
HallsofIvy said:Equivalently, write
[tex]\frac{f(x)- f(2)}{\sqrt{x}- \sqrt{2}}= \frac{f(x)- f(2)}{x- 2}\frac{x- 2}{\sqrt{x}- \sqrt{2}}[/tex]
[tex]= \frac{f(x)- f(2)}{x- 2}\left(\frac{1}{\frac{\sqrt{x}- \sqrt{2}}{x- 2}}\right)
\text{ and think about the derivatives of f(2) and }\sqrt{x}[/tex] at x= 2.
A limit is a mathematical concept that represents the value a function approaches as the input approaches a certain value, often denoted by the symbol "x" or "a". It is used to describe the behavior of a function near a specific point.
The derivative of a function at a point is defined as the slope of the tangent line to the graph of the function at that point. This slope can also be thought of as the rate of change of the function at that point. The limit of the function as the input approaches that point is equal to the derivative of the function at that point.
Yes, it is possible for a function to have a limit at a point but not have a derivative at that point. This occurs when the function is not continuous at that point, meaning that there is a gap or jump in the graph of the function at that point.
To find the limit of a function as x approaches a non-zero number, you can simply plug in the value of the number into the function and evaluate. If the resulting value is undefined, you can use algebraic techniques such as factoring or rationalizing the denominator to simplify the expression and find the limit.
The relationship between limits and derivatives is crucial in understanding the behavior of a function near a specific point. It allows us to determine the slope or rate of change of a function at a point, which has many real-world applications in fields such as physics, engineering, and economics. Additionally, it provides a foundation for more advanced concepts in calculus and other branches of mathematics.