Clifland

2021-01-10

Use the method of your choice to evaluate the following limits.

$\underset{(x,y)\to (1,1)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{2{x}^{2}-xy-{y}^{2}}$

Dora

Skilled2021-01-11Added 98 answers

Given:

$\underset{(x,y)\to (1,1)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{2{x}^{2}-xy-{y}^{2}}$

To evaluate:

The given limit.

Solution:

Here,

$\underset{(x,y)\to (1,1)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{2{x}^{2}-xy-{y}^{2}}$

On simplifying the equation,

The required equation is,

$\underset{(x,y)\to (1,1)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{{x}^{2}-xy2-{y}^{2}}=\underset{(x,y)\to (1,1)}{lim}\frac{x+2y}{2x+y}$

Plug in the values (x,y)=(1,1)

$\underset{(x,y)\to (1,1)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{{x}^{2}-xy2-{y}^{2}}=\frac{1+2\times 1}{2\times 1+1}$

$\underset{(x,y)\to (1,1)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{{x}^{2}-xy2-{y}^{2}}=\frac{1+2}{2+1}$

$\underset{(x,y)\to (1,1)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{{x}^{2}-xy2-{y}^{2}}=\frac{3}{3}$

$\underset{(x,y)\to (1,1)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{{x}^{2}-xy2-{y}^{2}}=1$

Which is required.

To evaluate:

The given limit.

Solution:

Here,

On simplifying the equation,

The required equation is,

Plug in the values (x,y)=(1,1)

Which is required.

Jeffrey Jordon

Expert2022-04-01Added 2575 answers