Answer:
a = -1/2
Step-by-step explanation:
You want the value of 'a' that will make (√(1+ax) -√(1-ax))/x approach -1/2 as x approaches zero.
Limit
At x=0, the function is defined as (2·0 +1)/(0 -2) = -1/2. In order for the function to be continuous there, the limit of the rational expression
[tex]\dfrac{\sqrt{1+ax}-\sqrt{1-ax}}{x}[/tex]
must be -1/2 as x approaches 0 from the left.
Solve for a
We can set the value of this to -1/2 and solve for the value of 'a', recognizing that we're interested in the value of 'a' as x approaches zero.
[tex]\dfrac{\sqrt{1+ax}-\sqrt{1-ax}}{x}=-\dfrac{1}{2}\\\\\\(\sqrt{1+ax}-\sqrt{1-ax})=-\dfrac{x}{2}\\\\\\(\sqrt{1+ax}-\sqrt{1-ax})(\sqrt{1+ax}+\sqrt{1-ax})=-\dfrac{x}{2}(\sqrt{1+ax}+\sqrt{1-ax})\\\\(1+ax)-(1-ax)=-\dfrac{x}{2}(\sqrt{1+ax}+\sqrt{1-ax})\\\\4a=-(\sqrt{1+ax}+\sqrt{1-ax})\qquad\text{multiply by $\dfrac{2}{x}$}\\\\4a=-2\qquad\text{limit as $x\to 0$}\\\\a=-\dfrac{1}{2}\qquad\text{divide by 4}[/tex]
The value of 'a' that makes the function continuous is -1/2.
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Additional comment
Obviously, we didn't need to multiply and divide by x. We could have multiplied by the sum of the roots and simplified the expression with x in place on the left:
((1+ax) -(1 -ax))/x = -1/2(√( ) + √( )) ⇒ 2a = -1 when x→0
