Clifland

2021-01-10

Use the method of your choice to evaluate the following limits.
$\underset{\left(x,y\right)\to \left(1,1\right)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{2{x}^{2}-xy-{y}^{2}}$

Dora

Given:
$\underset{\left(x,y\right)\to \left(1,1\right)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{2{x}^{2}-xy-{y}^{2}}$
To evaluate:
The given limit.
Solution:
Here,
$\underset{\left(x,y\right)\to \left(1,1\right)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{2{x}^{2}-xy-{y}^{2}}$
On simplifying the equation,
The required equation is,
$\underset{\left(x,y\right)\to \left(1,1\right)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{{x}^{2}-xy2-{y}^{2}}=\underset{\left(x,y\right)\to \left(1,1\right)}{lim}\frac{x+2y}{2x+y}$
Plug in the values (x,y)=(1,1)
$\underset{\left(x,y\right)\to \left(1,1\right)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{{x}^{2}-xy2-{y}^{2}}=\frac{1+2×1}{2×1+1}$
$\underset{\left(x,y\right)\to \left(1,1\right)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{{x}^{2}-xy2-{y}^{2}}=\frac{1+2}{2+1}$
$\underset{\left(x,y\right)\to \left(1,1\right)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{{x}^{2}-xy2-{y}^{2}}=\frac{3}{3}$
$\underset{\left(x,y\right)\to \left(1,1\right)}{lim}\frac{{x}^{2}+xy-2{y}^{2}}{{x}^{2}-xy2-{y}^{2}}=1$
Which is required.

Jeffrey Jordon