Does $\lim_{(x,y)\rightarrow (0,0)}\frac{(x^2-y^2) \cos (x+y)}{x^2+y^2}$ exists?
Answer
Writing the limit in polar coordinates (i.e. $x=r \cos \theta$ and $y=r \sin \theta$) we get
\[\lim_{(x,y) \rightarrow (0,0)}\frac{(x^2-y^2)\cos(x+y)}{x^2+y^2}=\lim_{r \rightarrow 0} \frac{r^2(\cos^2 \theta-\sin^2 \theta) \cos(r(\cos \theta +\sin \theta))}{r^2}\]
\[=\lim_{r \rightarrow 0} (\cos^2 \theta-\sin^2 \theta) \cos(r(\cos \theta +\sin \theta))\]
\[=\cos^2 \theta-\sin^2 \theta. \]
Since the above depends on $\theta$ the limit is question does not exists. Indeed on the line $y=x$ we have $\theta=\frac{\pi}{4}$
\[\cos^2 \theta-\sin^2 \theta =(\frac{\sqrt{2}}{2})^2-(\frac{\sqrt{2}}{2})^2=0.\]
On the line $y=0$ we have $\theta=0$, and hence
\[\cos^2 \theta-\sin^2 \theta =1^2-0^2=1.\]
Thus the limit does not exist, since if it exists it must be unique.
- answered
- 399 views
- $2.00
Related Questions
- Calculus - stationary points, Taylor's series, double integrals..
-
The given equation is x² - 2mx + 2m - 1=0
Determine m. - Find the coordinates of the point $(1,1,1)$ in Spherical coordinates
- Equation from Test
- Finding the arc length of a path between two points
- I need help with the attched problem about definite integrals
- Line Integral
- Stokes' theorem $\int_S \nabla \times F \cdot dS= \int_C F\cdot dr$ verification