# 9.2: Limits and Continuity

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1. Evaluate $$\displaystyle \lim_{(x, y) \rightarrow (0, 0)} (2x^2 - 3y + 7)$$, if possible. If the limit does not exist, explain why.

2. 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.

3. 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.

4. Evaluate $$\displaystyle \lim_{(x, y) \rightarrow (0, 0)} \dfrac{xy}{x^2 + y^2}$$, if possible. If the limit does not exist, explain why.

5. 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.

6. 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.

7. Evaluate $$\displaystyle \lim_{(x, y) \rightarrow (0, 0)} \dfrac{xy}{x^2 + y^2}$$, if possible. If the limit does not exist, explain why.

8. Evaluate $$\displaystyle \lim_{(x, y) \rightarrow (2, 0)} \dfrac{x^2 \sin y}{y}$$, if possible. If the limit does not exist, explain why.

9. 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.

10. 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 $$y = x^2$$.]

11. 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.

12. Evaluate $$\displaystyle \lim_{(x, y) \rightarrow (0, 0)} \dfrac{x^3 + y^3}{x^2 + y^2}$$ by converting to polar coordinates.

13. Evaluate $$\displaystyle \lim_{(x, y) \rightarrow (0, 0)} x \ln (x^2 + y^2)$$ by converting to polar coordinates.

14. Evaluate $$\displaystyle \lim_{(x, y, z) \rightarrow (0, 0, 0)} \dfrac{xyz}{x^2 + y^2 + z^2}$$ by converting to spherical coordinates.

15. Determine the region in the $$xy$$-plane over which $$f(x, y) = \sin(xy)$$ is continuous.

16. Determine the region in the $$xy$$-plane over which $$g(x, y) = \ln(x^2 + y^2)$$ is continuous.

17. Determine the region in the $$xy$$-plane over which $$h(x, y) = \dfrac{1}{xy}$$ is continuous.

9.2: Limits and Continuity is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.