(a.) Find limx?0 - f(x) and limx?0 + f(x). (Find limit as x approaches to 0 from
ID: 2846133 • Letter: #
Question
(a.) Find limx?0- f(x) and limx?0+ f(x). (Find limit as x approaches to 0 from left and right)?
[Hint: Rewrite f(x) by multiplying by the conjugate of the numerator to both the numerator and the denominator.]
(b.) Sketch the graph of y = f(x) in the viewing window [ -2; 2] [ -1; 1].
(c.) Use the graph to check your answer to part (a.) Explain any interesting behavior, particularly involving signs.
Thanks.
Explanation / Answer
As the hint suggests, multiply the top and bottom by the conjugate of the numerator. That is:
[dfrac{sqrt{2 - sqrt{4 - x^2}}}{x} cdot dfrac{sqrt{2 + sqrt{4 - x^2}}}{sqrt{2 + sqrt{4 - x^2}}} = dfrac{sqrt{4 - (4 - x^2)}}{xsqrt{2 + sqrt{4 - x^2}}} = dfrac{sqrt{x^2}}{xsqrt{2 + sqrt{4 - x^2}}} = dfrac{|x|}{xsqrt{2 + sqrt{4 - x^2}}}]
We then need to evaluate the limits. Observe that for both side-limits, the signs of the function are different.
[lim_{x ightarrow 0^-} dfrac{-x}{xsqrt{2 + sqrt{4 - x^2}}} = lim_{x ightarrow 0^-} -dfrac{1}{sqrt{2 + sqrt{4 - x^2}}} = -dfrac{1}{2}]
[lim_{x ightarrow 0^+} dfrac{x}{xsqrt{2 + sqrt{4 - x^2}}} = lim_{x ightarrow 0^+} dfrac{1}{sqrt{2 + sqrt{4 - x^2}}} = dfrac{1}{2}]
So both side limits are not equal to each other.
Note: (sqrt{x^2} = |x|), not (sqrt{x^2} = x). The value of (sqrt{x^2}) depends on the value of (x). If (x geq 0), then (sqrt{x^2} = x). Otherwise, (sqrt{x^2} = -x).
Check the link of the graph of (f(x)) within the given intervals: http://www.wolframalpha.com/input/?i=y+%3D+%28%E2%88%9A%282+-+%E2%88%9A%284+-+x%5E2%29%29%29%2Fx+for+-2+%3C+x+%3C+2%2C+-1+%3C+y+%3C+1
(f(x)) is negative at (x < 0), whereas (f(x)) is positive at (x > 0)
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