Question

Here is limit question\[\lim_{x \rightarrow 0}\left( \frac{ (9^{x}-1)(4^{x}-1) }{ \sqrt{2}-\sqrt{1+\cos(x)}} \right)\]

Answer 1

i hate this type of limit questions

Answer 2

divide numerator and denom. by x^2

Answer 3

lim (a^x-1)/x=ln a

Answer 4

multiply and divide by conjugate :
\(\sqrt2+\sqrt{1+cos x}\)

Answer 5

not that way @hartnn

Answer 6

in denominator, 1-cos x/x^2 remains

Answer 7

which is half,
and u had ln9.ln4.
final answer ln9.ln/root 2 ??

Answer 8

ln9.ln4/root 2 ??

Answer 9

nope

Answer 10

Let me give you the answer first\[8sqrt{2}(ln 3)(ln 2) \]

Answer 11

L'Hopital's ?

Answer 12

nope

Answer 13

ln9.ln4/root 2 is exactly same as 8sqrt2(ln3)(ln2)

Answer 14

i was correct, just needed to simplify

Answer 15

you are not @ hartnn

Answer 16

ln9=2ln 3
ln4 =2ln2
ln9.ln4/root 2 = 4ln 2.ln3 / root 2 = 8 ln2.ln3 *root2
got it ?

Answer 17

nopes, my expression is exactly simplified to your answer.

Answer 18

you are incorrect while you rationalize sqrt 2

Answer 19

can't find how 8 comes......yes, i m getting 2root2 instead of 8root2

Answer 20

then...

Answer 21

let me write it properly again.

Answer 22

\(\huge\frac{4ln3.ln2*2\sqrt2}{1/2}\)
i made the mistake of writing 2root 2 in denom. actually its in numerator.
now i m getting 16

Answer 23

I'm getting 16 instead of 8 lol

Answer 24

@hartnn yea..

Answer 25

Right, Here is how you solve it @hartnn

Answer 26

me too getting 16

Answer 27

\[\lim_{x \rightarrow 0}\left( \frac{ (9^{x}-1)(4^{x}-1) }{ \sqrt{2}-\sqrt{1+\cos(x)}} \right)\]
\[\lim_{x \rightarrow 0}\left( \frac{ (9^{x}-1)(4^{x}-1)times(sqrt{2}+sqrt{1+cos(x)}) }{ 2-1-cos(x)} \right) \]
\[\lim_{x \rightarrow 0}\left( \frac{ (9^{x}-1)(4^{x}-1)(sqrt{2}+sqrt{1+cos(x)}) }{ 1-sqrt{1-sin^{2}x }} \right) \]
\[\lim_{x \rightarrow 0}( \frac{ (9^{x}-1)(4^{x}-1)(sqrt{2}+sqrt{1+cos(x)})times(1+sqrt{1-sin^{2}x)} }{ sin^{2}x } ) \]
\[\lim_{x \rightarrow 0}( (\frac{ x }{ sin(x) })^{2}times(\frac{ 9^{x}-1 }{ x })times(\frac{ 4^{x}-1 }{ x })times(sqrt{2}+sqrt{1+cos(x)})times(1+sqrt{1-sin^{2}x}))\]
\[ (1)^{2}times(ln 9)times(ln 4)times(2sqrt{2})times(1) \]
\[8sqrt{2}(ln 3)(ln 2) \]

Answer 28

I am closing this

Answer 29

wait how u getting 8, it should be 16

Answer 30

See the steps

Answer 31

u forgot one of the '2's
for 1+sqrt(1-sin^2 x)

Answer 32

Ya,ya ...I forget it thanks it 16.

Answer 33

u wrote it as times 1

Answer 34

it should be times 2

Answer 35

It is true