9.2: Limits and Continuity
- Page ID
- 144342
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- Evaluate \(\displaystyle \lim_{(x, y) \rightarrow (0, 0)} (2x^2 - 3y + 7)\), if possible. If the limit does not exist, explain why.
- Evaluate \(\displaystyle \lim_{(x, y) \rightarrow (2, -1)} \dfrac{x^2}{x^2 + y^2}\), if possible. If the limit does not exist, explain why.
- Evaluate \(\displaystyle \lim_{(x, y) \rightarrow (0, 0)} \dfrac{x^2}{x^2 + y^2}\), if possible. If the limit does not exist, explain why.
- Evaluate \(\displaystyle \lim_{(x, y) \rightarrow (0, 0)} \dfrac{xy}{x^2 + y^2}\), if possible. If the limit does not exist, explain why.
- Evaluate \(\displaystyle \lim_{(x, y) \rightarrow (0, 0)} \dfrac{x^4 - y^4}{x^2 + y^2}\), if possible. If the limit does not exist, explain why.
- Evaluate \(\displaystyle \lim_{(x, y) \rightarrow (2, 1)} \dfrac{x - y - 1}{\sqrt{x - y} - 1}\), if possible. If the limit does not exist, explain why.
- Evaluate \(\displaystyle \lim_{(x, y) \rightarrow (0, 0)} \dfrac{xy}{x^2 + y^2}\), if possible. If the limit does not exist, explain why.
- Evaluate \(\displaystyle \lim_{(x, y) \rightarrow (2, 0)} \dfrac{x^2 \sin y}{y}\), if possible. If the limit does not exist, explain why.
- Evaluate \(\displaystyle \lim_{(x, y) \rightarrow (0, 0)} \dfrac{x^2 + \sin^2 y}{2x^2 + y^2}\), if possible. If the limit does not exist, explain why.
- Evaluate \(\displaystyle \lim_{(x, y) \rightarrow (0, 0)} \dfrac{x^2y}{x^4 + y^2}\), if possible. If the limit does not exist, explain why. [Hint: In addition to other paths, consider the limit along the path \(y = x^2\).]
- Evaluate \(\displaystyle \lim_{(x, y, z) \rightarrow (0, 0, 0)} \dfrac{x^2 - y^2 - z^2}{x^2 + y^2 - z^2}\), if possible. If the limit does not exist, explain why.
- Evaluate \(\displaystyle \lim_{(x, y) \rightarrow (0, 0)} \dfrac{x^3 + y^3}{x^2 + y^2}\) by converting to polar coordinates.
- Evaluate \(\displaystyle \lim_{(x, y) \rightarrow (0, 0)} x \ln (x^2 + y^2)\) by converting to polar coordinates.
- Evaluate \(\displaystyle \lim_{(x, y, z) \rightarrow (0, 0, 0)} \dfrac{xyz}{x^2 + y^2 + z^2}\) by converting to spherical coordinates.
- Determine the region in the \(xy\)-plane over which \(f(x, y) = \sin(xy)\) is continuous.
- Determine the region in the \(xy\)-plane over which \(g(x, y) = \ln(x^2 + y^2)\) is continuous.
- Determine the region in the \(xy\)-plane over which \(h(x, y) = \dfrac{1}{xy}\) is continuous.