Finding a&b in a Limit Problem

acreech

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Jun 13, 2010
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lim ((ax+b)[sup:k3zjht91]1/2[/sup:k3zjht91]-2)/x=1
x-0

Find numbers a and b


I don't know where to begin with this problem, is there a formula or something that can be used? Thanks for your help.
 
Rationalize the numerator. That gives ax+b-4 in the numerator.

a and b are the same value.
 
Given: limx0(ax+b)1/22x = 1, find a and b.\displaystyle Given: \ \lim_{x\to0}\frac{(ax+b)^{1/2}-2}{x} \ = \ 1, \ find \ a \ and \ b.

Hence, by inspection, we must have the indeterminate form 00 and then we\displaystyle Hence, \ by \ inspection, \ we \ must \ have \ the \ indeterminate \ form \ \frac{0}{0} \ and \ then \ we

must have Dx[(ax+b)1/22] = 1.\displaystyle must \ have \ D_x[(ax+b)^{1/2}-2] \ = \ 1.

Ergo, (ax+b)1/22 = 0      (ax+b) = 4\displaystyle Ergo, \ (ax+b)^{1/2}-2 \ = \ 0 \ \implies \ (ax+b) \ = \ 4

Dx[(ax+b)1/22] = a2(ax+b)1/2 = 1      ax+b = a24.\displaystyle D_x[(ax+b)^{1/2}-2] \ = \ \frac{a}{2(ax+b)^{1/2}} \ = \ 1 \ \implies \ ax+b \ = \ \frac{a^{2}}{4}.

a24 = 4      a2 = 16, a = ±4\displaystyle \frac{a^2}{4} \ = \ 4 \ \implies \ a^{2} \ = \ 16, \ a \ = \ \pm4

Use 4, then (4x+b)1/2 = 2      4x+b = 4, 4(0)+b = 4, b = 4\displaystyle Use \ 4, \ then \ (4x+b)^{1/2} \ = \ 2 \ \implies \ 4x+b \ = \ 4, \ 4(0) +b \ = \ 4, \ b \ = \ 4

Therefore, the limx0(ax+b)1/22x = 1     limx0(4x+4)1/22x = 1.\displaystyle Therefore, \ the \ \lim_{x\to0}\frac{(ax+b)^{1/2}-2}{x} \ = \ 1 \ \implies \lim_{x\to0}\frac{(4x+4)^{1/2}-2}{x} \ = \ 1.

Hence, a = b = 4, QED\displaystyle Hence, \ a \ = \ b \ = \ 4, \ QED
 
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