Chapter 11: Q21E (page 632)
Determine the set of points at which the function is continuous.
\(f(x,y) = \frac{{1 + {x^2} + {y^2}}}{{1 - {x^2} - {y^2}}}\)
The function f(x, y) is continuous in its domain, i.e. ,
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Use polar coordinates to find the limit. If \((r,\theta )\) are polar coordinates of the point \((x, y)\) with \(r \ge 0\) note that \(r \to {0^ + }\) as \((x,y) \to (0,0)\)
\(\mathop {lim}\limits_{(x,y) \to (0,0)} \left( {{x^2} + {y^2}} \right)\ln \left( {{x^2} + {y^2}} \right)\)
Find the value of \(\frac{{\partial T}}{{\partial p}},\frac{{\partial T}}{{\partial q}}\) and\(\frac{{\partial T}}{{\partial r}}\),using the chain rule if \(T = \frac{v}{{2u + v}},u = pq\sqrt r \) and \(v = p\sqrt q r\)when \(p = 2,q = 1\) and \(r = 4.\)
Use a computer graph of the function to explain why the limit does not exist.
\(\mathop {\lim }\limits_{(x,y) \to (0,0)} \frac{{2{x^2} + 3xy + 4{y^2}}}{{3{x^2} + 5{y^2}}}\)
Suppose that the equation \(F(x,y,z) = 0\) implicitly defines each of the three variables \(x,y\), and \(z\) as functions of the other two: \(z = f(x,y),y = g(x,z),x = h(y,z)\). If \(F\) is differentiable and \({F_s},{F_y}\), and \({F_z}\) are all nonzero, show that
\(\frac{{\partial z}}{{\partial x}}\frac{{\partial x}}{{\partial y}}\frac{{\partial y}}{{\partial z}} = - 1\).
Find the limit, if it exists, or show that the limit does not exist.
\(\mathop {lim}\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{5{y^4}co{s^2}x}}{{{x^4} + {y^4}}}\)
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