1. **Problem Statement:** Investigate the existence of limits for various functions as $(x,y) \to (0,0)$ using path tests and polar coordinates, analyze continuity, and interpret geometric and physical meanings. --- ### Q1: Investigate limits by checking different paths (a) $$\lim_{(x,y)\to(0,0)} \frac{x^2 y}{x^4 + y^2}$$ - Try path $y=0$: $$\frac{x^2 \cdot 0}{x^4 + 0} = 0$$ - Try path $y = x^2$: $$\frac{x^2 \cdot x^2}{x^4 + x^4} = \frac{x^4}{2x^4} = \frac{1}{2}$$ - Different limits along different paths $\Rightarrow$ limit does not exist. (b) $$\lim_{(x,y)\to(0,0)} \frac{xy}{x^2 - y^2}$$ - Path $y=mx$: $$\frac{x(mx)}{x^2 - (mx)^2} = \frac{m x^2}{x^2 - m^2 x^2} = \frac{m}{1 - m^2}$$ - Limit depends on $m$, so limit does not exist. (c) $$\lim_{(x,y)\to(0,0)} \frac{x^3}{x^2 + y^2}$$ - Path $y=0$: $$\frac{x^3}{x^2} = x \to 0$$ - Path $y = x$: $$\frac{x^3}{x^2 + x^2} = \frac{x^3}{2x^2} = \frac{x}{2} \to 0$$ - All paths tested tend to 0, so limit likely exists and equals 0. --- ### Q2: Use polar coordinates to determine limits Recall: $x = r \cos\theta$, $y = r \sin\theta$, and $r \to 0$. (a) $$\lim_{r\to0} \frac{x^2 + y^2}{\sqrt{x^2 + y^2}} = \lim_{r\to0} \frac{r^2}{r} = \lim_{r\to0} r = 0$$ (b) $$\lim_{r\to0} \frac{x^2 y^2}{(x^2 + y^2)^{3/2}} = \lim_{r\to0} \frac{r^4 \cos^2\theta \sin^2\theta}{r^3} = \lim_{r\to0} r \cos^2\theta \sin^2\theta = 0$$ (c) $$\lim_{r\to0} \frac{x^2 - y^2}{\sqrt{x^2 + y^2}} = \lim_{r\to0} \frac{r^2 (\cos^2\theta - \sin^2\theta)}{r} = \lim_{r\to0} r (\cos^2\theta - \sin^2\theta) = 0$$ All limits are 0 but depend on $\theta$ in the factor, so polar method suggests limit 0. --- ### Q3: Function $f(x,y) = \frac{x^2 y}{x^4 + y^2}$ (a) Path test shows different limits (from Q1a), so limit does not exist. (b) Polar coordinates: $$f(r,\theta) = \frac{r^2 \cos^2\theta \cdot r \sin\theta}{r^4 \cos^4\theta + r^2 \sin^2\theta} = \frac{r^3 \cos^2\theta \sin\theta}{r^2 (r^2 \cos^4\theta + \sin^2\theta)} = \frac{r \cos^2\theta \sin\theta}{r^2 \cos^4\theta + \sin^2\theta}$$ As $r \to 0$, denominator $\to \sin^2\theta$, so limit depends on $\theta$ and does not exist. (c) Both methods agree: limit depends on path/angle, so no unique limit. --- ### Q4: Continuity at origin (a) $$f(x,y) = \begin{cases} \frac{x^2 y}{x^2 + y^2}, & (x,y) \neq (0,0) \\ 0, & (0,0) \end{cases}$$ - From Q3, limit does not exist at origin, so not continuous. (b) $$f(x,y) = \begin{cases} \frac{xy}{x^2 + y^2}, & (x,y) \neq (0,0) \\ 0, & (0,0) \end{cases}$$ - Limit depends on path (Q1b), so not continuous. (c) $$f(x,y) = \begin{cases} \frac{x^2 - y^2}{x^2 + y^2}, & (x,y) \neq (0,0) \\ 0, & (0,0) \end{cases}$$ - Limit depends on $\theta$ (Q2c), so not continuous. --- ### Q5: Find $k$ for continuity (a) For continuity, limit must equal $k$. - From Q3 and Q4a, limit does not exist, so no $k$ makes it continuous. (b) $$f(x,y) = \begin{cases} \frac{x^4 + y^4}{x^2 + y^2}, & (x,y) \neq (0,0) \\ k, & (0,0) \end{cases}$$ - Polar: $$\frac{r^4 (\cos^4\theta + \sin^4\theta)}{r^2} = r^2 (\cos^4\theta + \sin^4\theta) \to 0$$ - Limit is 0, so $k=0$ for continuity. --- ### Q6: Conceptual (a) Example: $$f(x) = \begin{cases} 0, & x \neq 0 \\ 1, & x=0 \end{cases}$$ limit at 0 is 0 but function value is 1, so not continuous. (b) Example: $$f(x,y) = \frac{x^2 y}{x^4 + y^2}$$ all directional limits along lines exist but limit does not exist (Q1a). (c) Checking only $y=mx$ misses nonlinear paths; limit may differ on other curves. --- ### Q7: Surface $f(x,y) = \frac{x^2 - y^2}{x^2 + y^2}$ (a) Near origin, surface varies between -1 and 1 depending on $\theta$. (b) Along $y=x$: $$f = \frac{x^2 - x^2}{x^2 + x^2} = 0$$ (c) Along $y=0$: $$f = \frac{x^2 - 0}{x^2 + 0} = 1$$ (d) Level curves: $\frac{x^2 - y^2}{x^2 + y^2} = c$ for $c \in (-1,1)$ are hyperbolas. --- ### Q8: Physical Interpretation of $T(x,y) = \frac{x^2 y}{x^2 + y^2}$ (a) Temperature limit at origin does not exist (Q1a). (b) Model is not physically stable near origin due to non-unique limit. (c) Geometrically, temperature surface has different values approaching origin from different directions. --- **Final answers:** - Limits in Q1a,b do not exist; Q1c limit is 0. - Polar method confirms limits in Q2. - Continuity fails in Q4a,b,c. - $k=0$ in Q5b; no $k$ in Q5a. - Conceptual examples in Q6. - Surface behavior and level curves in Q7. - Physical interpretation in Q8.