Let-f-x-5x-40-x-2-1-Calculate-the-derivative-of-f-4-without-using-the-answer-of-question-2-2-Calculate-the-derivative-of-f-x-3-Calculate-lim-x-f-x-All-your-answe

Question Number 52542 by hassentimol last updated on 09/Jan/19

Let f (x) = \frac{5 x + 40}{x + 2}

(1) Calculate the derivative of f (4) without

using the answer of question (2),

(2) Calculate the derivative of f (x),

(3) Calculate \lim_{x \to + \infty} f (x) .

All your answers should be correctly proved and detailed .

C a n y o u p l e a s e h e l p m e f o r t h a t e x e r c i s e .

Commented by Abdo msup. last updated on 10/Jan/19

1) f^{^{'}} (4) = {l i m}_{h \to 0} \frac{f (4 + h) - f (4)}{h}

= {l i m}_{h \to 0} \frac{\frac{5 (4 + h) + 40}{4 + h + 2} - 10}{h}

= {l i m}_{h \to 0} \frac{60 + 5 h - 60 - 10 h}{h (6 + h)}

= {l i m}_{h \to 0} \frac{- 5 h}{h (6 + h)} = {l i m}_{h \to 0} \frac{- 5}{6 + h} = - \frac{5}{6}

2) f (x) = \frac{5 (x + 2) + 30}{x + 2} = 5 + \frac{30}{x + 2)} \Rightarrow

f^{^{'}} (x) = - \frac{30}{{(x + 2)}^{2}} f o r x \neq 2 .

Commented by Abdo msup. last updated on 10/Jan/19

3) {l i m}_{x \to + \infty} f (x) = {l i m}_{x \to + \infty} \frac{5 x}{x} = 5 .

Commented by hassentimol last updated on 10/Jan/19

Thanks sir!!

Answered by tanmay.chaudhury50@gmail.com last updated on 09/Jan/19

2) \frac{d f (x)}{d x} = \frac{(x + 2) \frac{d (5 x + 40)}{d x} - (5 x + 40) \frac{d}{d x} (x + 2)}{{(x + 2)}^{2}}

= \frac{(x + 2) (5) - (5 x + 40) (1)}{{(x + 2)}^{2}}

= \frac{5 x + 10 - 5 x - 40}{{(x + 2)}^{2}}

= \frac{- 30}{{(x + 2)}^{2}}

3) \lim_{x \to \infty} \frac{5 x + 40}{x + 2}

= \lim_{x \to \infty} \frac{5 + \frac{40}{x}}{1 + \frac{2}{x}}

= \frac{5 + 0}{1 + 0} = 5

1) f (x) = \frac{5 x + 40}{x + 2}

{(\frac{d f}{d x})}_{a t x = 4} = \lim_{h \to 0} \frac{f (4 + h) - f (4)}{h}

= \lim_{h \to 0} \frac{\frac{5 (4 + h) + 40}{(4 + h) + 2} - \frac{60}{6}}{h}

= \lim_{h \to 0} \frac{20 + 5 h + 40 - 40 - 10 h - 20}{h (6 + h) 6}

= \lim_{h \to 0} \frac{- 5 h}{h (36 + 6 h)} = \frac{- 5}{36}

Commented by hassentimol last updated on 09/Jan/19

T h a n k y o u s i r! G o d b l e s s y o u

T h a n k y o u v e r y m u c h, I^{'} m n o w u n d e r s t a n d i n g .

Commented by tanmay.chaudhury50@gmail.com last updated on 09/Jan/19

m o s t w e l c o m e \dots