Finding a LimitDate: 02/28/2002 at 22:47:04 From: Corrie Subject: Limits My calculus teacher gave my class the answer to this limit; however, we cannot figure out how to get the answer that is given. If you could help me with a proof, I would be very grateful. 1 - sqrt(2x^2 - 1) lim ------------------ x->1 x - 1 Thanks! Date: 03/01/2002 at 08:47:05 From: Doctor Peterson Subject: Re: Limits Hi, Corrie. If we just replace x with 1, we get (1-1)/(1-1), which we can't evaluate; so we'd like to rearrange the expression so we can cancel something out. The big thing that is in our way is the "1-sqrt" in the numerator; how about if we "rationalize the numerator" by multiplying by the conjugate? Then (for x not equal to 1): 1 - sqrt(2x^2 - 1) 1 + sqrt(2x^2 - 1) ------------------ * ------------------ = x - 1 1 + sqrt(2x^2 - 1) 1 - (2x^2 - 1) 2(1 - x^2) --------------------------- = --------------------------- (x - 1)(1 + sqrt(2x^2 - 1)) (x - 1)(1 + sqrt(2x^2 - 1)) Can you see the next step? Factor the numerator, cancel, and you will have an expression that can be evaluated at x = 0, and which is equal to the original expression for all other x, so that its value at x = 0 is the limit you want. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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