Find the limit.
-$\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \sin \left( \frac { x ^ { 10 } + y ^ { 5 } } { x - y + 10 } \right)$
A) $\frac { 1 } { 10 }$
B) 1
C) 0
D) No limit

Question

Quizplus · Accepted Answer

Using the Squeeze Theorem, we can show that the limit is 0.

Let $f(x,y) = \sin \left( \frac { x ^ { 10 } + y ^ { 5 } } { x - y + 10 } ight)$. Note that $-1 \leq \sin(	heta) \leq 1$ for all $	heta$.

Now consider the function $g(x,y) = \frac{x^{10}+y^5}{x-y+10}$. We will find two functions $h_1(x,y) \leq g(x,y) \leq h_2(x,y)$ that both have limit 0 as $(x,y) 	o (0,0)$.

First, we can simplify $g(x,y)$ by dividing both the numerator and denominator by $x$: 
$$g(x,y) = \frac{x^{9}+y^5/x^5}{1-y/x+10/x}.$$ 
Since $\lim_{(x,y) 	o (0,0)} y/x = 0$, we have 
$$\lim_{(x,y) 	o (0,0)} (1-y/x+10/x) = 1.$$ 
Furthermore, we can find constants $M_1$ and $M_2$ such that 
\begin{align*} 
|h_1(x,y)| &= \left| \frac{y^5}{x-y+10} ight| \leq \frac{y^5}{|x-y| - 10} \
&\leq \frac{y^5}{\sqrt{x^2+y^2}} \leq \frac{1}{\sqrt{2}}y^{\frac{11}{2}} \qquad 	ext{(using the AM-GM inequality)}.
\end{align*}
Thus, $\lim_{(x,y) 	o (0,0)} h_1(x,y) = 0$. Similarly, we have 
\begin{align*} 
|h_2(x,y)| &= \left| \frac{x^{9}+y^5/x^5}{x-y+10} ight| \leq |x^{9}| + |y^5/x^5| \cdot \left| \frac{1}{x-y+10} ight| \
&\leq |x^{9}| + y^5 \cdot \frac{1}{|x-y| + 10} \
&\leq x^2 + y^2 + y^5 \cdot \frac{1}{|x-y| + 10} \qquad 	ext{(using the inequality $x^9 \leq x^2$ when $|x| \leq 1$)}.
\end{align*}
Note that the denominator $|x-y| + 10$ is bounded below by a constant when $(x,y)$ is close to $(0,0)$, so the last term in the above inequality tends to 0 as $(x,y) 	o (0,0)$. Thus, $\lim_{(x,y) 	o (0,0)} h_2(x,y) = 0$.

Finally, using the Squeeze Theorem, we have 
$$\lim_{(x,y) 	o (0,0)} f(x,y) = \lim_{(x,y) 	o (0,0)} \sin \left( \frac { x ^ { 10 } + y ^ { 5 } } { x - y + 10 } ight) = 0.$$

Find the Limit lim⁡(x,y)→(0,0)sin⁡(x10+y5x−y+10)\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \sin \left( \frac { x ^ { 10 } + y ^ { 5 } } { x - y + 10 } \right)lim(x,y)→(0,0)​sin(x−y+10x10+y5​)

Find the Limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \sin \left( \frac { x ^ { 10 } + y ^ { 5 } } { x - y + 10 } \right)$