Try the fastest way to create flashcards
Question

# Evaluate the limits or explain why the limit fails to exist.$\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { ( x + y ) ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$

Solution

Verified
Step 1
1 of 2

To prove that the limit does not exist we can rearrange the given terms using the properties of limit.

$\lim_{\left(x,y\right)\to\left(0,0\right)}\dfrac{\left(x+y\right)^2}{x^2+y^2}=\lim_{\left(x,y\right)\to\left(0,0\right)}\dfrac{x^2+y^2}{x^2+y^2}+\lim_{\left(x,y\right)\to\left(0,0\right)}\dfrac{2xy}{x^2+y^2}$

The first term is obviously equal to $1$. For the second term take the paths $x=t$ and $y=2t$ and let $t=0$.

We now have:

$\lim_{t\to 0}\dfrac{4t^2}{5t^2}=\dfrac{4}{5}$

However if we set $x=t$ and take $y=3t$ we get a different result:

$\lim_{t\to 0}\dfrac{6t^2}{10t^2}=\dfrac{3}{5}$

Therefore limit does not exist.

## Recommended textbook solutions

#### Thomas' Calculus

14th EditionISBN: 9780134438986 (11 more)Christopher E Heil, Joel R. Hass, Maurice D. Weir
10,142 solutions

#### Vector Calculus

4th EditionISBN: 9780321780652 (3 more)Susan J. Colley
1,866 solutions

#### Calculus: Early Transcendentals

8th EditionISBN: 9781285741550 (6 more)James Stewart
11,085 solutions

#### Calculus: Early Transcendentals

9th EditionISBN: 9781337613927 (1 more)Daniel K. Clegg, James Stewart, Saleem Watson
11,050 solutions