Question

Evaluate the limit
lim (√(9+4x^(2)))/(11+10x)
x->INF

Answer 1

\[\lim_{x\to\infty}{\sqrt{9+4x^2}\over11+10x^2}\]correct?

Answer 2

(11+10x)

Answer 3

\[\lim_{x\to\infty}{\sqrt{9+4x^2}\over11+10x}\]

Answer 4

yup

Answer 5

much easier...
divide top and bottom by x, then take the limit

Answer 6

so its √(9+4x^)/(11+10)?

Answer 7

√(9+4x)/(11+10)?

Answer 8

you must be more careful with the numerator

Answer 9

what is\[{\sqrt{9+4x^2}\over x}\]simplified?

Answer 10

9+4x?

Answer 11

1) where did the square root go?
2) how does the square root sign affect trying to divide by x?

Answer 12

the answers:
1) the square root must stay
2) in order to divide this by x that means we must square it, to get it into the radical

Answer 13

square the numerator

Answer 14

?

Answer 15

9+4x^2

Answer 16

you have to square x to put it into the radical, just as it works the other way....

Answer 17

\[\sqrt{\frac a{x^2}}=\frac1x\sqrt{a}\]so it works the other way too

Answer 18

\[{\sqrt a\over x}=\sqrt {a\over{x^2}}\]

Answer 19

so we have\[{\sqrt{9+4x^2}\over x}=\sqrt{{9+4x^2\over x^2}}=\sqrt{\frac9{x^2}+4}\]for the numerator

Answer 20

why did you square x?

Answer 21

it's perhaps best seen by example...

Answer 22

\[{\sqrt
{16}\over2}=\sqrt{\frac{16}{2^2}}=\sqrt{\frac{16}4}=2\]\[{\sqrt
{25}\over5}=\sqrt{\frac{25}{5^2}}=\sqrt{\frac{16}4}=1\]

Answer 23

etc.

Answer 24

try it with uglier numbers if you like, I don't want to type it all out

Answer 25

in principle it works the same as simplifying\[\sqrt{24}\]if you know how to do that:\[\sqrt{24}=\sqrt4\cdot6=2\sqrt6\]

Answer 26

only here we are doing it with division

Answer 27

\[\sqrt{\frac{50}4}=\frac{\sqrt{50}}2=\frac{2\sqrt5}2=\sqrt5 \]

Answer 28

I'm sorry, but I'm a bit too tired to go into a more detailed explanation. I hope you can meditate on this and decipher it.

Answer 29

yup i got it thank you

Answer 30

well what I wrote above was wrong, should have been:\[\sqrt{\frac{50}4}=\frac{\sqrt{50}}2=\frac{5\sqrt2}2\]told you I'm tired.
later!

Answer 31

hhhh thanks

Answer 32

2/10 got it