suppose $f(x,y)=\frac{|x|^a|y|^b}{|x|^c+|y|^d}$ and $a,b,c,d &gt;0$ how would you show that if $\frac{a}{c} + \frac{b}{d} &gt;1$ then $\lim\limits_{(x,y)\to(0,0)}\frac{|x|^a|y|^b}{|x|^c+|y|^d}$ exists and equals $0$?  I've been trying to use the squeeze theorem and set up an inequality but im really struggling.
85888
Ted Shifrin on this site posed an interesting problem to me: show that   $$ \lim_{(x,y) \to (0,0)} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \frac\alpha\gamma + \frac\beta\delta &gt; 1, \,\,\,\,\text{where } \alpha, \beta, \gamma, \delta &gt;0$$  I think I've got the $(\Leftarrow)$ direction as follows: assume WLOG that $\gamma \leq \delta$. Then switch to polar coordinates and get   $$\frac {r^ {\alpha + \beta} (*) } {r^\gamma (1(*) + r^{\delta - \gamma}(*))},$$  where $(*)$ represents some trig stuff that is bounded near zero.   Edit: I need to make sure that $(1(*) + r^{\delta - \gamma}(*))$ is bounded below here.  The other direction is giving me some trouble. It seems we need a lower bound for the fraction (something to force to zero), and I'm not sure where to find one. In particular, I'm not sure what to do with the denominator. Any ideas?
71836
Let $a, b, c, d &gt; 0$. Prove that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq a + b + c + d$.   I'm supposed to prove this by AM-GM, but I can't see how. Any help would be appreciated.
99590
Given $$f(x,y) = \frac{9xy}{x^2 + y^2},$$ determine the limit of $f(x,y)$ as $(x,y) \to (2,1)$. The answer for this would be $\dfrac{18}{5}$ right?
76345
Let $f(x,y)$ be a function of two variables. Given that  $$\frac{\partial f}{\partial x}=0$$ for all $y$, and  $$\frac{\partial f}{\partial y}=0$$ for all $x$, is there a theorem that states $f(x,y)=$ constant for all $x$, $y$? Or how would I prove this rigorously?
76460
I need to use this theorem as part of a proof. Basically it goes as follows:  If $f(x_0) &gt; g(x_0)$ and $\frac{d}{dx} f(x) &gt; \frac{d}{dx} g(x)$ for all $x \geq x_0$, then $f(x) &gt; g(x)$ for all $x\geq x_0$.  What's the name of this theorem, if it has one? Or if it doesn't, is there a simple way to prove it that I could include in my proof? Thanks.
43414
Right now, I'm doing a question:   $$\lim_{(x,y)\to(1,0)}\frac{xy-y}{(x-1)^2 +y^2}$$  I know the limit doesn't exist, but I can't figure out how to prove it. I tried putting $x=1$, and getting $0/y^2$, and put $y=0$, got $0/(x^2-2x+1)$, but I don't think that does it.  (edit: this is not a duplicate; I'm having a hard time getting good explanations, hence why I asked)
59372
How to find limit as $x \to 0$ of $x\sin(1/x)$?    For $x^2\sin(1/x)$, I know it's $0$ since by the Squeeze theorem, $-x^2 \le x^2\sin(1/x) \le x^2$, but for $x\sin(1/x)$, I run into some problems when applying Squeeze theorem.
110601
$$\lim\limits_{x\to 1}\frac{\sqrt x-x^2 }{1-\sqrt x}$$  I'm not entirely sure how to go about this, since rationalizing anything doesn't quite work for me. Just looking for some help.
65397
Let $x&gt;0$. Find $$\lim_{n\to\infty}\frac{n^{x}}{(1 + x)^{n}}.$$   What I was thinking involved an inequality, by reason of considering the fact that  $$(1 + x)^{n} &gt; C(n, k) x^{k}$$ for some $k$ such that $n&gt;k$. Expanding the binomial coefficient and taking the $n$ common from each bracket, we have something like $$(1 + x)^{n} &gt; C(n, k) x^{n} &gt; \frac {x^{k}n^{k}}{k! 2^{k}}.$$ But now, I am totally clueless. By intuition, I know that the limit has to be zero, so I just want to prove using the concept of inequalities that what I thought was the right approach (being the sandwich theorem). Any help would be appreciated!
85884
Show that for $x,y,z &gt; 0$ the inequality is true: $\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}+x+y+z \geq \frac{(x+y)^2}{y+z}+\frac{(y+z)^2}{z+x}+\frac{(z+x)^2}{x+y}$  I have tried Holder, but i had no luck. Please give me a hint of how to start.
39655
I want to prove that $\lim_{(x,y)\to (0,0)} \frac{\sin(x+y)}{x+y} = 1$  Is it sufficient to say that if $u = x+y$ then as $(x,y) \rightarrow (0,0)$ then $u \rightarrow 0$ and then evaluate $\lim_{u\to 0} \frac{\sin u}{u}$ which is 1 by L'Hôpital's rule?  I feel that this is somehow "cheating" and I would really like to prove it in a more rigorous way, i.e., by using an $\epsilon$ - $\delta$ argument.  I have tried for a long time to use the one variable version and prove that if $0 \lt \sqrt{x^2 + y^2} \lt \delta$ then somehow $0 \lt |x+y| \lt \delta$ and it would follow from the one variable case that $| \frac{\sin(x+y)}{x+y} - 1| \lt \epsilon$. However, because $|x+y| = 0$ when $y = -x$ I am not getting anywhere with this.  Any input on this would be helpful :)
108666
Prove that   $$\lim_{x\to \frac{\pi}{2}} \frac{1}{\big(x-\frac{\pi}{2}\big)}+{\tan(x)}=0.$$   I'm not really sure how to proceed. I know that I should not try L'Hôpital's rule (tried that) but not sure how I would incorporate into the Squeeze Theorem or how I would use continuity.  Thanks!  Edit: Turns out I was really dumb and you do use L'Hôpital's rule twice. I made the mistake of differentiating the whole quotient rather than the function on top and the bottom of the vinculum separately.
54311
I want to determine $$\lim\limits_{(x,y) \to (0,0)} \frac{x^2·y^3}{x^4+y^6}$$  I'm sure the limit exists, it's zero because I tried to find other different limits in line and parabola points ($(x,mx)$ and $(x,mx^2)$) but every time I got zero. So, I have to use the definition or sequences to confirm.
40101
Find the limit: $$\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$$   I have no idea what to do, but I thought that this is the limit of the derivative of $f(x)=\left(1+x\right)^{1/x}$, as $x$ tends to 0. Any help?
36661
I'm trying to prove the general      $$ \lim_{n\to\infty} \frac{an+c}{bn+d} = \frac{a}{b} $$   using the definition of the limit of a sequence, for any $\varepsilon &gt; 0 \,\, \exists N \in \mathbb{N}$ s.t. for every $n \geq N$ $|a_n - a| &lt; \varepsilon$. My (not so valiant) efforts:  \begin{align*}  \left| \frac{an+c}{bn+d} - \frac{a}{b} \right| &amp;&lt; \varepsilon \\ \left| \frac{abn + bc - abn - ad}{b^2n+bd} \right| &amp;&lt; \varepsilon \\ \left| \frac{bc-ad}{bn+d} \right| &amp;&lt; \varepsilon \left| b \right| \\ \left| bn+d \right| &amp;&gt; \left| \frac{bc - ad}{\varepsilon b} \right| \\ \left| bn+d \right| &amp;&gt; \left| \frac{c}{\varepsilon} - \frac{ad}{\varepsilon b} \right| \end{align*}  I'm used to being able to manipulate constants or remove absolute values when solving these kinds of problems, but with general constants I'm not sure what to do next.
51546
Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$  I'm trying to solve it by using the squeeze theorem but I'm stuck.  I'm looking for a function $g(x)$ such that $g(x) \leq x\lfloor\frac{1}{x}\rfloor \leq x(\frac{1}{x}) = 1$  Can anybody give me a hint?
22998
Find multivariable limit of: $$\lim_{ \left( x,y\right) \rightarrow \left(0,0 \right)}\frac{x^2y}{x^2+y^3}$$ How to find that limit? I was trying to do the following, but i am not able to find a proper inequality: $$| \frac{x^2y}{x^2+y^3} | = |y-\frac{y^4}{x^2+y^3}| \le$$