lim-x-7-x-a-4-x-7-b-find-a-b-

Question Number 146505 by mathdanisur last updated on 13/Jul/21

\underset{x \to 7}{l i m} \frac{\sqrt{x - a} - 4}{x - 7} = b f i n d a \cdot b = ?

Answered by ajfour last updated on 13/Jul/21

i j u s t d u n n o b u t l e t s e e w h t i t s

b o u t . .

i f l i m i t h a s t o e x i s t, i . e .

b f i n i t e t h e n d^{r} \to 0 \Rightarrow n^{r} \to 0

s o x - a = 16

b u t x \to 7 \Rightarrow a \to - 9

b u t t h e n

b = \frac{\frac{\partial n^{r}}{\partial x}}{\frac{\partial d^{r}}{\partial x}} = \frac{\frac{1}{2 \sqrt{x - a}}}{1}

b u t g a i n x \to 7, s o

\Rightarrow b = \frac{1}{2 \sqrt{7 - (- 9)}} = \frac{1}{8}

n o w i f y o u a s k

a b f o r g e t (\cdot), u h a v e - \frac{9}{8}

Commented by mathdanisur last updated on 13/Jul/21

t h a n k s S e r

Answered by mathmax by abdo last updated on 13/Jul/21

f (x) = \frac{\sqrt{x - a} - 4}{x - 7} changement x - 7 = t give

f (x) = \frac{\sqrt{t + 7 - a} - 4}{t} = g (t) (t \to 0) \Rightarrow

g (t) = \frac{\sqrt{7 - a} \sqrt{1 + \frac{t}{7 - a}} - 4}{t} \sim \frac{\sqrt{7 - a} \times (1 + \frac{t}{2 (7 - a)}) - 4}{t}

\frac{\sqrt{7 - a} - 4}{t} + \frac{1}{2 \sqrt{7 - a}} = b \Rightarrow \sqrt{7 - a} - 4 = 0 \Rightarrow 7 - a = 16 \Rightarrow a = - 9 \Rightarrow

b = \frac{1}{2 \sqrt{7 + 9}} = \frac{1}{8} \Rightarrow ab = - \frac{9}{8}

Commented by mathdanisur last updated on 13/Jul/21

t h a n k y o u S e r