Use the method of your choice to evaluate the following limits. lim_{(x,y)rightarrow(1,1)}frac{x^2+xy-2y^2}{2x^2-xy-y^2}

Use the method of your choice to evaluate the following limits. lim_{(x,y)rightarrow(1,1)}frac{x^2+xy-2y^2}{2x^2-xy-y^2}

Question
Limits and continuity
asked 2021-01-10
Use the method of your choice to evaluate the following limits.
\(\lim_{(x,y)\rightarrow(1,1)}\frac{x^2+xy-2y^2}{2x^2-xy-y^2}\)

Answers (1)

2021-01-11
Given:
\(\lim_{(x,y)\rightarrow(1,1)}\frac{x^2+xy-2y^2}{2x^2-xy-y^2}\)
To evaluate:
The given limit.
Solution:
Here,
\(\lim_{(x,y)\rightarrow(1,1)}\frac{x^2+xy-2y^2}{2x^2-xy-y^2}\)
On simplifying the equation,
The required equation is,
\(\lim_{(x,y)\rightarrow(1,1)}\frac{x^2+xy-2y^2}{x^2-xy2-y^2}=\lim_{(x,y)\rightarrow(1,1)}\frac{x+2y}{2x+y}\)
Plug in the values (x,y)=(1,1)
\(\lim_{(x,y)\rightarrow(1,1)}\frac{x^2+xy-2y^2}{x^2-xy2-y^2}=\frac{1+2\times1}{2\times1+1}\)
\(\lim_{(x,y)\rightarrow(1,1)}\frac{x^2+xy-2y^2}{x^2-xy2-y^2}=\frac{1+2}{2+1}\)
\(\lim_{(x,y)\rightarrow(1,1)}\frac{x^2+xy-2y^2}{x^2-xy2-y^2}=\frac{3}{3}\)
\(\lim_{(x,y)\rightarrow(1,1)}\frac{x^2+xy-2y^2}{x^2-xy2-y^2}=1\)
Which is required.
0

Relevant Questions

asked 2020-11-27
Use the method of your choice to evaluate the following limits.
\(\lim_{(x,y)\rightarrow(0,\pi/2)}\frac{1-\cos xy}{4x^2y^3}\)
asked 2021-03-06
Use the method of your choice to evaluate the following limits.
\(\lim_{(x,y)\rightarrow(2,0)}\frac{1-\cos y}{xy^2}\)
asked 2021-02-09
Use the method of your choice to evaluate the following limits.
\(\lim_{(x,y)\rightarrow(1,0)}\frac{\sin xy}{xy}\)
asked 2021-05-12
Evaluate the following limits.
\(\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{x^{2}+xy-xz-yz}{x-z}\)
asked 2021-03-08
Use the method of your choice to evaluate the following limits.
\(\lim_{(x,y)\rightarrow(4,0)}x^2y\ln xy\)
asked 2021-02-21
Use the method of your choice to evaluate the following limits.
\(\lim_{(x,y)\rightarrow(1,0)}\frac{y\ln y}{x}\)
asked 2021-01-02
Use the method of your choice to evaluate the following limits.
\(\lim_{(x,y)\rightarrow(0,1)}\frac{y\sin x}{x(y+1)}\)
asked 2021-05-27
Evaluate \(\lim_{x \rightarrow \infty} \frac{\sin h x}{e^x}\)
asked 2021-03-09
Evaluate the following limits. \(\lim_{(x,y)\rightarrow(2,2)}\frac{y^2-4}{xy-2x}\)
asked 2020-11-01
Evaluate the following limits.
\(\lim_{(x,y)\rightarrow(2,-1)}(xy^8-3x^2y^3)\)
...