MHB How do you calculate limits in two variables?

  • Thread starter Thread starter mathmari
  • Start date Start date
  • Tags Tags
    Limits
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hey! :o

I have to calculate the following limits, if they exists.
  1. $$\lim_{(x, y) \rightarrow (0, 0)} (x^2+y^2+3)$$
  2. $$\lim_{(x, y) \rightarrow (0, 0)} \frac{xy}{x^2+y^2+2}$$
  3. $$\lim_{(x, y) \rightarrow (0, 0)} \frac{e^xy}{x+1}$$
  4. $$\lim_{(x, y) \rightarrow (0, 0)} \frac{\cos x-1-x^2/2}{x^4+y^4}$$
  5. $$\lim_{(x, y) \rightarrow (0, 0)} \frac{(x-y)^2}{x^2+y^2}$$

I have done the following:

  1. $x^2+y^2+3$ is continuous for each $(x, y) \in \mathbb{R}^2$, so we have that $\lim_{(x, y) \rightarrow (0, 0)} (x^2+y^2+3)=0^2+0^2+3=3$
  2. $\frac{xy}{x^2+y^2+2}$ is continuous for each $(x, y) \in \mathbb{R}^2$, so we have that $\lim_{(x, y) \rightarrow (0, 0)} \frac{xy}{x^2+y^2+2}=\frac{0 \cdot 0}{0^2+0^2+2}=2$
  3. $\frac{e^xy}{x+1}$ is continuous for $(x, y) \in \mathbb{R}^2 \setminus \{(-1, y)\}$, so we have that $\lim_{(x, y) \rightarrow (0, 0)} \frac{e^xy}{x+1}=\frac{e^0 0}{0+1}=0$

Is ths correct?? (Wondering) Could I improve something at the formulation?? (Wondering)

What could we do at the limits $4-5$ ?? (Wondering)
 
Physics news on Phys.org
mathmari said:
Hey! :o

I have to calculate the following limits, if they exists.
  1. $$\lim_{(x, y) \rightarrow (0, 0)} (x^2+y^2+3)$$
  2. $$\lim_{(x, y) \rightarrow (0, 0)} \frac{xy}{x^2+y^2+2}$$
  3. $$\lim_{(x, y) \rightarrow (0, 0)} \frac{e^xy}{x+1}$$
  4. $$\lim_{(x, y) \rightarrow (0, 0)} \frac{\cos x-1-x^2/2}{x^4+y^4}$$
  5. $$\lim_{(x, y) \rightarrow (0, 0)} \frac{(x-y)^2}{x^2+y^2}$$

I have done the following:

  1. $x^2+y^2+3$ is continuous for each $(x, y) \in \mathbb{R}^2$, so we have that $\lim_{(x, y) \rightarrow (0, 0)} (x^2+y^2+3)=0^2+0^2+3=3$
  2. $\frac{xy}{x^2+y^2+2}$ is continuous for each $(x, y) \in \mathbb{R}^2$, so we have that $\lim_{(x, y) \rightarrow (0, 0)} \frac{xy}{x^2+y^2+2}=\frac{0 \cdot 0}{0^2+0^2+2}=2$
  3. $\frac{e^xy}{x+1}$ is continuous for $(x, y) \in \mathbb{R}^2 \setminus \{(-1, y)\}$, so we have that $\lim_{(x, y) \rightarrow (0, 0)} \frac{e^xy}{x+1}=\frac{e^0 0}{0+1}=0$

Is ths correct?? (Wondering) Could I improve something at the formulation?? (Wondering)

What could we do at the limits $4-5$ ?? (Wondering)

For 2. last time I checked, 0/2 = 0, not 2...
 
mathmari said:
Hey! :o

I have to calculate the following limits, if they exists.
  1. $$\lim_{(x, y) \rightarrow (0, 0)} (x^2+y^2+3)$$
  2. $$\lim_{(x, y) \rightarrow (0, 0)} \frac{xy}{x^2+y^2+2}$$
  3. $$\lim_{(x, y) \rightarrow (0, 0)} \frac{e^xy}{x+1}$$
  4. $$\lim_{(x, y) \rightarrow (0, 0)} \frac{\cos x-1-x^2/2}{x^4+y^4}$$
  5. $$\lim_{(x, y) \rightarrow (0, 0)} \frac{(x-y)^2}{x^2+y^2}$$

I have done the following:

  1. $x^2+y^2+3$ is continuous for each $(x, y) \in \mathbb{R}^2$, so we have that $\lim_{(x, y) \rightarrow (0, 0)} (x^2+y^2+3)=0^2+0^2+3=3$
  2. $\frac{xy}{x^2+y^2+2}$ is continuous for each $(x, y) \in \mathbb{R}^2$, so we have that $\lim_{(x, y) \rightarrow (0, 0)} \frac{xy}{x^2+y^2+2}=\frac{0 \cdot 0}{0^2+0^2+2}=2$
  3. $\frac{e^xy}{x+1}$ is continuous for $(x, y) \in \mathbb{R}^2 \setminus \{(-1, y)\}$, so we have that $\lim_{(x, y) \rightarrow (0, 0)} \frac{e^xy}{x+1}=\frac{e^0 0}{0+1}=0$

Is ths correct?? (Wondering) Could I improve something at the formulation?? (Wondering)

What could we do at the limits $4-5$ ?? (Wondering)

5. is just screaming to be converted to polars...

$\displaystyle \begin{align*} \lim_{(x,y) \to (0,0)} \frac{(x-y)^2}{x^2+y^2} &= \lim_{r\to 0}\frac{\left[r\cos{(\theta )} - r\sin{(\theta )} \right] ^2}{r^2} \\ &= \lim_{r \to 0} \frac{r^2 \left[ \cos{(\theta )} - \sin{(\theta )} \right] ^2}{r^2} \\ &= \lim_{r \to 0} \left[ \cos{(\theta )} - \sin{(\theta )} \right] ^2 \\ &= \left[ \cos{(\theta )} - \sin{(\theta )} \right] ^2 \end{align*}$

Since this value changes according to the angle approached on, that means different paths yield different values, and thus the limit does not exist.
 
For the first three one could we formulate it as followed?? (Wondering)

  1. $(x, y) \rightarrow (0, 0)$
    $x \rightarrow 0$ and $y \rightarrow 0$
    So, $x^2+y^2+3 \rightarrow 0^2+0^2+3=3$
  2. $(x, y) \rightarrow (0, 0)$
    $x \rightarrow 0$ and $y \rightarrow 0$
    So, $\frac{xy}{x^2+y^2+2} \rightarrow \frac{0 \cdot 0}{0^2+0^2+2}=0$
  3. $(x, y) \rightarrow (0, 0)$
    $x \rightarrow 0$ and $y \rightarrow 0$
    So, $\frac{e^xy}{x+1} \rightarrow \frac{e^0 \cdot 0}{0+1}=0$

What could we do at the limit $4.$ to check if it exists?? (Wondering)
 
mathmari said:
For the first three one could we formulate it as followed?? (Wondering)

  1. $(x, y) \rightarrow (0, 0)$
    $x \rightarrow 0$ and $y \rightarrow 0$
    So, $x^2+y^2+3 \rightarrow 0^2+0^2+3=3$
  2. $(x, y) \rightarrow (0, 0)$
    $x \rightarrow 0$ and $y \rightarrow 0$
    So, $\frac{xy}{x^2+y^2+2} \rightarrow \frac{0 \cdot 0}{0^2+0^2+2}=0$
  3. $(x, y) \rightarrow (0, 0)$
    $x \rightarrow 0$ and $y \rightarrow 0$
    So, $\frac{e^xy}{x+1} \rightarrow \frac{e^0 \cdot 0}{0+1}=0$

What could we do at the limit $4.$ to check if it exists?? (Wondering)

Hey mathmari! (Mmm)

What you had in the OP seemed fine to me.
This also looks fine. (Nod)

For 4 and 5, I'd check what happens if you approach the origin along different lines, say $y=0$ and $x=0$. (Thinking)
 
Ok... Thanks a lot! (Mmm)
 
Back
Top