The limit below is a definition of f'(a). Determine the function f(x) and the value of a. lim_{h...

Question:

The limit below is a definition of {eq}f'(a) {/eq}. Determine the function {eq}f(x) {/eq} and the value of {eq}a {/eq}.

{eq}\displaystyle \lim_{h \to 0} \dfrac {\dfrac 1 {(2 + h)} - 0.5} {h} {/eq}.

Limit Definition Of A Derivative:

The limit definition of a derivative is used to find the derivative of any type of function. So it can be used to derive any formula which is present in the list of rules of derivatives. It states that the derivative of a function {eq}f(x) {/eq} at a value {eq}x=a {/eq} is:

$$f'(a)=\lim _{h \rightarrow 0} \dfrac{f(a+h)-f(a)}{h} $$

Answer and Explanation:

The limit definition of a derivative states that the derivative of a function {eq}f(x) {/eq} at a value {eq}x=a {/eq} is:

$$f'(a)=\lim _{h \rightarrow 0} \dfrac{f(a+h)-f(a)}{h} $$

Compare this limit with the gien limit:

$$\lim _{h \rightarrow 0} \dfrac{\dfrac{1}{(2+h)}-0.5}{h}= \lim _{h \rightarrow 0} \dfrac{\dfrac{1}{(2+h)}- \dfrac{1}{2}}{h} $$

Then we get:

$$\begin{align} &f(a+h) =\dfrac{1}{(2+h)} \\ & f(a) = \dfrac{1}{2} \end{align} $$

Then we get:

{eq}\boxed{\mathbf{f(x) = \dfrac{1}{x}; \,\,\, a=2.}} {/eq}


Learn more about this topic:

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Using Limits to Calculate the Derivative

from Math 104: Calculus

Chapter 9 / Lesson 1
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