lim-x-1-x-5-1-7-1-1-x-7-1-9-1-3-

Question Number 67031 by hmamarques1994@gmai.com last updated on 21/Aug/19

\lim_{x \to - 1} \sqrt[3]{\frac{\sqrt[7]{x^{5}} + 1}{1 + \sqrt[9]{x^{7}}}} = ?

Commented by Tony Lin last updated on 22/Aug/19

{(\lim_{x \to - 1} \frac{\frac{5}{7} x^{\frac{- 2}{7}}}{\frac{7}{9} x^{\frac{- 2}{9}}})}^{\frac{1}{3}}

= {(\lim_{x \to - 1} \frac{45_{9} \sqrt{x^{2}}}{49_{7} \sqrt{x^{2}}})}^{\frac{1}{3}}

= {(\frac{45}{49})}^{\frac{1}{3}}

p l e a s e c h e c k

Commented by hmamarques1994@gmail.com last updated on 22/Aug/19

y e s \underset{}{!}

= \frac{\sqrt[3]{315}}{7} \approx 0, 972013

t h a n k y u!

Commented by mathmax by abdo last updated on 22/Aug/19

y o u h a v e u s e d {l i m}_{x \to x_{0}} {(\frac{f (x)}{g (x)})}^{r} = {({l i m}_{x \to x_{0}} \frac{f^{^{'}} (x)}{g^{^{'}} (x)})}^{r}

i s t h i s c o r r e c t ? (i t s n o t h o s p i t a l t h e o r e m \dots .)

Commented by Tony Lin last updated on 22/Aug/19

L^{'} H o s p i t a l r u l e :

i f \lim_{x \to x_{0}} f (x) = \lim_{x \to x_{0}} g (x) = 0 a n d \lim_{x \to x_{0}} \frac{f^{'} (x)}{g^{'} (x)} e x i s t s

\Rightarrow \lim_{x \to x_{0}} \frac{f (x)}{g (x)} = \lim_{x \to x_{0}} \frac{f^{'} (x)}{g^{'} (x)}

i f \lim_{x \to x_{0}} f (x) = \pm \infty, \lim_{x \to x_{0}} g (x) = \pm \infty a n d \lim_{x \to x_{0}} \frac{f^{'} (x)}{g^{'} (x)} e x i s t s

\Rightarrow \lim_{x \to x_{0}} \frac{f (x)}{g (x)} = \lim_{x \to x_{0}} \frac{f^{'} (x)}{g^{'} (x)}

i f \lim_{x \to x_{0}} f (x) e x i s t s

\lim_{x \to x_{0}} {[f (x)]}^{n} = {[\lim_{x \to x_{0}} f (x)]}^{n}

\Rightarrow P o w e r L a w o f L i m i t

o r w e c a n u s e C o m p o s i t i o n L a w t o k n o w

i f f i s c o n t i n u o u s a t \lim_{x \to x_{0}} g (x)

t h e n \lim_{x \to x_{0}} f (g (x)) = f (\lim_{x \to x_{0}} g (x))