Find-the-all-asymtotes-of-the-curve-y-5x-2-2x-1-x-2-

Question Number 123407 by bemath last updated on 25/Nov/20

F i n d t h e a l l a s y m t o t e s o f t h e

c u r v e y = \frac{5 x^{2} + 2 x - 1}{x - 2}

Commented by EVIMENEBASSEY last updated on 25/Nov/20

F i n d t h e a l l a s y m t o t e s o f t h e

c u r v e y = \frac{5 x^{2} + 2 x - 1}{x - 2}

solution

y (x - 2) = {5 x}^{2} + 2 x - 1

horizontal asymtote = 2

no horizontal asymtote .

but letting y = mx + c

(mx + c) (x - 2) = {5 x}^{2} + 2 x - 1

{mx}^{2} - mx + xc - 2 c = {5 x}^{2} + 2 x - 1

c = m = 0

caddet

Commented by liberty last updated on 25/Nov/20

(i) v e r t i c a l a s y m t o t e s : \lim_{x \to 2} y = \lim_{x \to 2} \frac{5 x^{2} + 2 x - 1}{x - 2} = \infty

w e g e t v e r t i c a l a s y m t o t e s x = 2

(i i) h o r i z o n t a l a s y m t o t e s : \lim_{x \to \pm \infty} y = \lim_{x \to \pm \infty} \frac{5 x^{2} + 2 x - 1}{x - 2}

c l e a r l y t h e l i m i t d o e s n o t e x i s t s o

t h e c u r v e n o h o r i z o n t a l a s y m t o t e s

(i i i) w e w a n t f i n d i n c l i n e d a s y m t o t e s

y = k x + b w h e r e {\begin{cases} k = \lim_{x \to \infty} \frac{y}{x} = \lim_{x \to \infty} \frac{5 x^{2} + 2 x - 1}{x^{2} - 2 x} = 5 \\ b = \lim_{x \to \infty} (y - k x) = \lim_{x \to \infty} (\frac{5 x^{2} + 2 x - 1}{x - 2} - 5 x) \end{cases}

\Rightarrow b = \lim_{x \to \infty} (\frac{5 x^{2} + 2 x - 1 - 5 x^{2} + 10 x}{x - 2})

\Rightarrow b = \lim_{x \to \infty} (\frac{12 x - 1}{x - 2}) = 12

t h u s s t r a i g h t l i n e y = 5 x + 12 i s

i n c l i n e d a s y m t o t e s

Commented by MJS_new last updated on 25/Nov/20

y = \frac{p (x)}{q (x)} with p, q are polynomes and degree (p) ⩾ degree (q)

you can always find existing asymptotic lines

or curves by dividing the fraction

y = \frac{7 x + 3}{x - 5} = 7 + \frac{38}{x - 5} \Rightarrow asymptotes {\begin{cases} x = 5 \\ y = 7 \end{cases}

y = \frac{7 x^{2} + 3}{x - 5} = 7 x + 35 + \frac{178}{x - 5} \Rightarrow asymptotes {\begin{cases} x = 5 \\ y = 7 x + 35 \end{cases}

y = \frac{7 x^{3} + 3}{x^{2} - 5} = 7 x + \frac{35 x + 3}{x^{2} - 5} \Rightarrow asymptotes {\begin{cases} x = \pm \sqrt{5} \\ y = 7 x \end{cases}

asymptotic curves :

y = \frac{7 x^{3} + 3}{x - 5} = 7 x^{2} + 35 x + 175 + \frac{878}{x - 5} \Rightarrow asymptotes {\begin{cases} x = 5 \\ y = 7 x^{2} + 35 x + 175 \end{cases}

y = \frac{7 x^{4} + 3}{x^{2} - 5} = 7 x^{2} + 35 + \frac{178}{x^{2} - 5} \Rightarrow asymptotes {\begin{cases} x = \pm \sqrt{5} \\ y = 7 x^{2} + 35 \end{cases}

\dots

Commented by bemath last updated on 25/Nov/20

Commented by bemath last updated on 25/Nov/20

Commented by MJS_new last updated on 25/Nov/20

for y = \frac{7 x^{4} + 3}{x^{2} - 5} and y = 7 x^{2} + 35 you will have to

set up a different window . try

- 4 ⩽ x ⩽ 4 and - 100 ⩽ y ⩽ 300

Commented by bemath last updated on 25/Nov/20

y e s s i r .

Answered by MJS_new last updated on 25/Nov/20

vertical asymptote x = 2

y = 5 x + 12 + \frac{23}{x - 2} \Rightarrow asymptote y = 5 x + 12

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