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Problem 58

Show that $$ \lim _{(x, y) \rightarrow(0,0)}\left[\left(2 x^{3}-y^{3}\right) /\left(x^{2}+y^{2}\right)=0\right. $$

Expert verified

In summary, to show that the given limit is 0, we first rewrote the function as \(f(x, y) = \frac{(2x - y)(x^2 + xy + y^2)}{x^2+y^2}\) and analyzed the limit as both x and y approach 0. We observed that both the numerator and the denominator tend to 0 when x and y approach 0. By using polar coordinates (x = rcosθ and y = rsinθ) and simplifying the limit, we see that as r → 0, the overall function will also tend to 0. Therefore, we have shown that the given limit is indeed \(0: \lim _{(x, y) \rightarrow(0,0)}\frac{2 x^{3}-y^{3}}{x^{2}+y^{2}}=0\).

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Chapter 3

Let \(\mathrm{f}: \mathrm{R}^{3} \rightarrow \mathrm{R}\) be given by $\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\left[\left(\mathrm{y}^{3} \mathrm{z}\right) /\left(1+\mathrm{x}^{2}+\mathrm{z}^{3}\right)\right]$ ] Show that \(\mathrm{f}\) is continuous at \((0,0,0)\).

Chapter 3

Show that the function \(\mathrm{f}: \mathrm{R}^{n} \rightarrow \mathrm{R}\) given by $\mathrm{f}\left(\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\right)=\max \left\\{\mathrm{x}_{1}, \ldots ., \mathrm{x}_{\mathrm{n}}\right\\}$ is continuous everywhere.

Chapter 3

To which class of continuous functions do the following functions belong?
a) Weierstrass' s nowhere differentiable function
b) $\mathrm{f}(\mathrm{x}, \mathrm{y},
z)=\mathrm{e}^{\mathrm{x}-\mathrm{y}+\mathrm{z}}$
c) \(f(t)=t^{3} e_{1}+(\sin t) e_{2}+t^{8 / 3} e_{3},-\infty

Chapter 3

Let \(\mathrm{f}: \mathrm{R}^{2} \rightarrow \mathrm{R}\) be a real valued function with domain \(\mathrm{R}^{2}\) which is of the form $$ f\left(x_{1}, x_{2}\right)=\left[\left(5 x_{1}\right) /\left(1+x^{2} 2\right)\right] $$ Show that \(\mathrm{f}\) is continuous at \((0,0)\).

Chapter 3

Let \(\mathrm{f}(\mathrm{x}, \mathrm{y})\) be given by $$ \begin{array}{cl} \mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{xy}^{2} /\left(\mathrm{x}^{2}+\mathrm{y}^{4}\right) & (\mathrm{x}, \mathrm{y}) \neq(0,0) \\ \text { and }=0 & \mathrm{x}=\mathrm{y}=0 \end{array} $$ Is f continuous at the origin?

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