Question

Evaluate the following limit:

\[sqrt{(9+11x^{2})} / (6+3x)]\

Answer 1

as x -> inf and as x-> -(inf)

Answer 2

apply La Hopitals rule

Answer 3

differentiate top, differentiate the bottom, and try again

Answer 4

y= ( 9 + 11x^2)^1/2

dy/dx = (1/2) ( 9 + 11x^2 )^-1/2 * 22x

Answer 5

= 11x / ( sqrt ( 9+11x^2) )

Answer 6

so we want to take the limit of\[\frac{ \frac{11x}{\sqrt{9+11x^2} } } { 3}\]

Answer 7

yea, that's what i got so far

Answer 8

applying the rule again because it is still indeterminate form "infinity/infinity"

Answer 9

so the (1/3) factor , bring that out

then we have

11 / [ 11x / sqrt(9+11x^2) ]

Answer 10

so that gives what

sqrt(9+11x^2) / x I think

Answer 11

\[\frac{\sqrt{9+11x^2}}{x}\]

Answer 12

sstill indeterminate , but if we apply the rule once more that x on the denominator will vanish

Answer 13

yeh I think something possibly went wrong, because it we differentiate that then we get back to

11x/ sqrt (9+11x^2)

Answer 14

I would just go with dominate term analysis here I think

Answer 15

I would just go with dominate term analysis here I think

Answer 16

when i apply the lospitl rule i don't have to do a quotien rule between the two func tion right?

Answer 17

i got the answers right thou thanks

to inf is sqrt11/3 and to -(inf) is -sqrt11/3

Answer 18

dominate term up the top is \[x \sqrt{11}\]

the dominate term on the bottom is x ( obviously )

Answer 19

so the quotient of these gives the answers sqrt(11) , and then you remember the factor of (1/3) we had earlier

Answer 20

how is it \[x sqrt {11}\]

Answer 21

do i just forget about the 9 ?

Answer 22

\[\lim_{x \rightarrow \infty}\frac{\sqrt{9+11x^{2}}}{6+3x}\]
\[ = \lim_{x \rightarrow \infty} \frac{x\sqrt{\frac{9}{x^2}+11}}{x(\frac{6}{x}+3)}\]

Answer 23

Becomes pretty easy from there.

Answer 24

Looks like \[\frac{\sqrt{11}}{3}\]

Answer 25

Did you follow that?

Answer 26

i didn't get how yu got sqrt11/3 from that. Did you just ignore everything that had an x ?

Answer 27

No. I took the limit as x goes to infinity. \(9/x^2\) will go to 0. 6/x will go to 0. The only things left will be the sqrt{11} and the 3 in the denominator.