Question
If $a, b, x,$ and $y$ are all positive and $a+b=1,$ evaluate (a) $\lim _{p \rightarrow 0} \frac{\ln \left(a x^{p}+b y^{p}\right)}{p}$ (b) $\lim _{p \rightarrow 0}\left(a x^{p}+b y^{p}\right)^{1 / p}$
If $a, b, x,$ and $y$ are all positive and $a+b=1,$ evaluate (a) $\lim _{p \rightarrow 0} \frac{\ln \left(a x^{p}+b y^{p}\right)}{p}$ (b) $\lim _{p \rightarrow 0}\left(a x^{p}+b y^{p}\right)^{1 / p}$
Answers
(a) Evaluate $ \lim_{x \to ^x} x \sin \frac {1}{x}. $
(b) Evaluate $ \lim_{x \to 0} x \sin \frac{1}{x}. $
(c) Illustrate parts (a) and (b) by graphing $ y = x \sin (1/x). $
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