Use graphing utility to graph function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods.
lim (sq root of (x+2) - sq root of 2) / x
x->0
Thanks!
12 answers
Dameon/Darly -- please use the same name for your posts.
http://www.wolframalpha.com/input/?i=%28%28x%2B2%29^.5-2^.5%29%2Fx
Thanks Damon!
how would i find the limit analytically?
x y
1.0 .318
0.5 .334
0.2 .345
0.1 .349
.05 .351
.02 .353
.01 .353
.001 .35351 etc
which looks much like (sqrt 2 )/4
[(x+2)^.5-2^.5]/x * [(x+2)^.5+2^.5]
-----------------------------------
[(x+2)^.5+2^.5]
[(x+2) -2]
= -------------
x [ (x+2)^.5+2^.5]
= 1/ [(x+2)^.5+2^.5]
which when x --->0 is
1/
1.0 .318
0.5 .334
0.2 .345
0.1 .349
.05 .351
.02 .353
.01 .353
.001 .35351 etc
which looks much like (sqrt 2 )/4
[(x+2)^.5-2^.5]/x * [(x+2)^.5+2^.5]
-----------------------------------
[(x+2)^.5+2^.5]
[(x+2) -2]
= -------------
x [ (x+2)^.5+2^.5]
= 1/ [(x+2)^.5+2^.5]
which when x --->0 is
1/
sorry
1/ [2 sqrt2)
multiply top and bottom by sqrt 2 to rationalize and we get
(sqrt 2) / 4
as we knew all along
1/ [2 sqrt2)
multiply top and bottom by sqrt 2 to rationalize and we get
(sqrt 2) / 4
as we knew all along
Now that was fun but you do the next one :)
is the answer 1/2?
oh nvm. thanks! next one?
no
it is one quarter of the square root of 2
it is one quarter of the square root of 2
sqrt 2/ 4 = .35355
which is what our table and graph said
which is what our table and graph said
oh okay. thanks!
1 answer