Question

Estimate the value of

lim as x approaches 0 ( x / [(sqrt of 1+3x) -1 ].
look below for the equation

Answer 1

\[\lim_{x \rightarrow 0} x/ \sqrt{1+3x} - 1\]

Answer 2

taht should be x divided by everything else

Answer 3

Use the limit laws to prove taht teh answer is correct

Answer 4

Since your limit ends up being 0/0, you can use L'Hopital's rule (assuming you have learned it)

using L'Hopital's looks like this:
lim x-> 0 x/(sqrt(1+3x)-1) = lim x-> 0 (dy/dx)/(d(sqrt(1+3x)-1)/dx

to make it more clear:\[1\div d(\sqrt{1+3x}-1)/dx\]
lim x -> 0 \[1/(1/2\sqrt{1+3x})(3)\]
so invert that and you get:
lim x -> 0\[2\sqrt{1+3x }/3\]
now, we take the limit and we are left with
2(1)/3
so your limit is 2/3

I know that may not have been very clear. I had a hard time making the steps look nice. let me know is you have a question

Answer 5

actually idk what the L-Hospital rule is adn im not sure what you did w/ the dy/dx

Answer 6

Hmmm, well, let me see what I can do without L'Hospital's rule, because that is the best way to do it here. give me a second.

Answer 7

no problem

Answer 8

so are you supposed to prove it exists or solve it?

Answer 9

the question as
a) estimate value of the function by graphing it
b) make a table of values fo x close to 0 and gues teh value of teh limit
c) use the limit laws to prove ur asnwer is correct

Answer 10

Hmm, do you know the squeeze theorem?

Answer 11

yeah

Answer 12

Well, I would try that. I haven't done squeeze theorem for limits in a while. but maybe:

use the equations:
y=x^2+2/3 for upper equation
y=-x^2 +2/3 for the lower equation?

I don't have a graphing calculator with me. but if you have one, maybe try graphing those three and see if it works.

Answer 13

well im actually not sure how to apply teh squeeze thereoem ...wen i graphed it both of the graphs are opposite.