Question 5: If we have two functions, f and g, is it always true thatg(x + h) – g(x)hー→0(f(x + h) + g(x + h)) – (f(x) + g(x)) ,f(x+ h) - f(x)limh-0h+ limlimh0hhWhatever your answer is, please provide your reasons and an example (i.e. two functions f and9, and if relevant, a specific value of x, that demonstrate your point).

We have to check that whether, limh→0fx+h-fxh+limh→0gx+h-gxh=limh→0fx+h+gx+h-fx+gxh is always true?

Answered: lim h0 S(x + h) – f(x) + lim h0 g(x +…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Step 1:

We have to check that whether,

$\begin{array}{rcl} \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} + \lim_{h \to 0} \frac{g (x + h) - g (x)}{h} & = & \lim_{h \to 0} [\frac{(f (x + h) + g (x + h)) - (f (x) + g (x))}{h}] \end{array}$

is always true?

Step 2:

$f (x) a n d g (x)$

Assume that the above function are differentiable.

$\frac{d [f (x)]}{d x} = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} a n d \frac{d [g (x)]}{d x} = \lim_{h \to 0} \frac{g (x + h) - g (x)}{h}$

Add the both terms to get,

$\begin{array}{rcl} \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} + \lim_{h \to 0} \frac{g (x + h) - g (x)}{h} & = & \lim_{h \to 0} [\frac{f (x + h) - f (x)}{h} + \frac{g (x + h) - g (x)}{h}] \\ = & \lim_{h \to 0} [\frac{f (x + h) - f (x) + g (x + h) - g (x)}{h}] \\ = & \lim_{h \to 0} [\frac{(f (x + h) + g (x + h)) - (f (x) + g (x))}{h}] \end{array}$

again $\frac{d [f (x) + g (x)]}{d x} = \begin{array}{rcl} \lim_{h \to 0} \end{array} \begin{array}{rcl} [\frac{(f (x + h) + g (x + h)) - (f (x) + g (x))}{h}] \end{array}$

Hence the given equation shows that

$\Rightarrow \begin{array}{rcl} \frac{d [f (x)]}{d x} + \frac{d [g (x)]}{d x} = \frac{d [f (x) + g (x)]}{d x} \end{array}$

Which is always true.

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