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First post! yay!
Okay so yesterday I had a problem set due and it had two definition of limit at infinity proofs on it. I ended up leaving them... well not blank but all I did was write the definition.
the question was:
prove the following statements using the appropriate definition of limit:
(i) Limit as x-> infinity sqrt(x) - x = -infinity
I got as far as:
Given N<0, there exists M>0 such that
x>M implies sqrt(x) - x < N
which I believe is the appropriate definition.
I always draw a blank on these proofs. The first thing I did when I made my account here was print the sticky on definition of limit proofs. I am going through it now.
Anyways any help on this would be amazing, Thanks!
Hey thanks for the help guys, sorry I haven't been able to follow up on my question, I had a bunch of assignments due on Monday. So I was able to prove this limit as follows:
I started playing with the inequality sqrt(x) - x < N. I multiplied by -1 to get x-sqrt(x)>-N --> sqrt(x)(sqrt(x) - 1) > -N. then since N<0 by assumption, --> -N>0.
Now if sqrt(x) - 1 > sqrt(-N) and sqrt(x) > sqrt(x) - 1 > sqrt(-N) then we get sqrt(x)(sqrt(x)-1) > sqrt(-N)sqrt(-N) = -N.
so I worked with sqrt(x) - 1 > sqrt(-N) ---> x > (1 + sqrt(-N))^2. So I concluded by choosing M = (1+sqrt(-N))^2 and showed that it works.
Can someone just confirm with me that this proof works? and also how do i post the math symbols like you guys did above me?